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due to the inequality of the balance-arms, and hence the ratio (W') : (w) will not represent the true capacity.

In those determinations where merely a ratio of weights is sought, as in the determination of the percentage of a constituent in a given substance, it is allowable to make direct weighings, if the substances which are weighed are always placed in the same pan, usually the left-hand one. By this procedure, the effect due to the inequality of the balance-arms is either eliminated or becomes negligible as shown in the following discussion.

For convenience, suppose, during the first weighing, the right arm to be longer than the left arm in the ratio R: L and let M be the mass of the substance used for analysis; suppose during the second weighing the ratio to be R': L' L L' and let m be the mass of the precipitate found. Then M and m will be R the corresponding apparent weights, namely, the weights as recorded. We see at once that

m

R'

L'

m R'

L
M
R

M

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or, in other words, provided that the ratio of the relative

lengths of the arms has been the same during the two weighings.

For a good balance having proper usage and care, the difference between L

L' and will be very small indeed in the case of consecutive weighings which R' R are made the one soon after the other. Under these conditions it follows that the ratio represented by the left-hand member of the above inequality approximates so very closely to the value of the right-hand member that for all purposes where direct weighings are allowable we can consider the two sides of the inequality as equal.

If the weighings are made on a faulty balance or if one of the balance-arms has been exposed to a higher temperature than the other through the weighing L' L

of a hot crucible or exposure to sunlight, the difference between and R may be large enough to invalidate our considering the ratio of the apparent weights as equal to the ratio of the masses.

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71. Gauss's Method of Double Weighing. If an object, whose true mass is M, is placed upon the left-hand balance-pan and is counterbalanced by weights W on the right-hand pan, we have by the principle of the lever-arm

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(2)

If now the object is placed in the right-hand pan and is counterbalanced by weights W' on the left-hand pan, we have

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Upon multiplying the first equation by the second and cancelling out the common factor RL, we get

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That is to say, the mass is equal to the square root of the product of the apparent weights.

For most purposes, however, it is sufficiently accurate to take the average of the apparent weights

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W+W'
2

(4)

if the two weights W and W' do not differ very much from each other. The validity of using the average is justified as follows. Let the difference between the apparent weights be a. Then W' = Wa according as W' is greater than or less than W. Whence from (3) we have

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Since this is an alternating series, the value of the remainder after n terms is numerically less than the value of the (n + 1) st term. If we include only the first two terms, W + then the value of the sum of all the succeeding

a2 8W

α 2'

2

α

terms is less than

That is to say, the value of M is represented by W+

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W + W'

W+W'

or

whence we establish the proposition that M is the same as
2
2

2

a2

except for a difference not greater than and since for our ordinary analytical

8W'

balances a is so very small in comparison with W, the difference is negligible. Determination of the ratio of the lengths of the balance-arms. If we divide

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the radical sign can be developed in terms of a series of powers of a and W, where as before a represents the difference between W' and W, i.e., W' Wa. Thus

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Discarding terms of a higher order than the first, we get

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the upper sign being used when W' is greater than W, and the lower sign being used when W' is less than W.

72. Borda's Method of Substitution. According to this method, the object, whose mass is M is placed in the right-hand pan of the balance and counterbalanced or tared by the necessary weight T, which may be lead shot, or preferably the weights of a duplicate set. Then the object is removed and the necessary weights put on in its place to counterbalance the counterpoise. We have from the first weighing

TL = MR

and from the second weighing

TL = ᎳᎡ

Dividing the first equation by the second, we see that

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As between the two methods, Gauss's and Borda's, it may be mentioned here that, other things being equal, the probable error of Gauss's method is less than that of Borda's. For problems involving the determination of only a few absolute weights, probably Gauss's method is to be preferred, but in the calibration of weights which involve quite a number of determinations, Borda's method lends itself far more readily to the problem.

73. Inaccuracy of the Weights. Since the individual weights of even a good set of weights often differ by as much as several tenths of a milligram from the denomination with which they are marked, it becomes necessary, as mentioned in § 64, to determine the values of the weights and construct a table accordingly. This operation, which is known as the calibration of weights, is described in detail in § 82.

The values of the weights can be determined, (1) in terms of the one gram weight of the set or (2) more preferably in terms of some standard weight which has been standardized by the Bureau of Standards at Washington.

If the values are determined according to (1), their use is limited to those determinations where the element of absolute value does not enter into the result in any way, or practically to those cases where the result can be obtained as the ratio of relative weights. Thus in the determination of the percentage of a constituent in a substance, since the percentage is one hundred times

the ratio of the weight of substance found to the weight of substance used, it is immaterial in what unit the two weights are expressed, provided that the same unit is used throughout. If the values are determined according to (2) they can be used in every case in place of (1) and furthermore in all cases where the element of absolute value does affect the final result. From what has just been said and from the fact that it is relatively a simple matter to obtain (2) from (1), it is obvious that the values under (2) are really the only ones that should be used in analytical work.

We have already given one illustration in § 70 where the element of absolute value enters into the determination. As another illustration, let us consider the preparation and use of standard solutions where it is desired to express the concentration of the reagent either in terms of molarity or normality. Since the former is the number of gram-formula weights, or moles, of the substance per liter of solution, while the latter is the number of gram equivalents of the substance per liter of solution, it is evident that the weights of the substance used must be known in terms of the standard gram or absolute unit, if a strict meaning is to attach to the compound units, molarity and normality.

74. Buoyant Effect of the Air. If an object is weighed first in air and then "in vacuo" with weights having a different density from the object, the values obtained in the two cases will not be the same. This difference is brought about by the fact that the buoyant effect of the air is greater upon the object or the weights according to which has the greater volume, because by the principle of Archimedes an object totally immersed in a fluid" is buoyed up by a force equal to the weight of the displaced fluid. In order to fix this idea with greater definiteness, let us consider the weighing of a liter of water with brass weights, first "in vacuo," then in air. Supposing the water to be at the temperature of 20°, its weight "in vacuo" will be 998.23 grams. If now the water is weighed in air, the air being at or near 20° and 760 mm. pressure, it will be found that the 998.23 grams are too heavy,

8

"By fluid is meant either liquid or gas.

9

It is assumed that the flask containing the water is tared by an exactly similar flask. See Table 4, § 99.

and the question arises, by how much? A moment's consideration shows us that the buoyant effect of the air upon the water is 1.20 grams because this is the weight of the liter of air displaced by the water; the buoyant effect of the air upon the brass weights is 0.15 gram because this is the weight of the 124.8 c.c. of air displaced by the brass weights.10 Therefore, the net amount by which the 998.23 grams are too heavy is 1.20 grams 0.15 gram 1.05 grams, so that we have finally as the weight in air of one liter of water, under the conditions named, the value 998.23 grams 1.05 grams 997.18 grams.

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Strictly speaking, the weight (mass) of an object is its weight "in vacuo" and where absolute values are being determined, or where the element of absolute values enters into a ratio, it is necessary to refer weighings to this standard. For most analytical purposes, however, where the intention is to obtain the result in terms of percentage, the ratio of the weights in air, as far as solids are concerned, will give a value which is sensibly the same as that which would be given by the weights "in vacuo."

In regard to liquids it depends largely upon the purpose of the weighing as to whether weights "in vacuo" must be used or whether weights in air are allowable. As already pointed out the weights "in vacuo" must be used in determining, for purposes of calibration, the amount of water contained or delivered by a piece of volumetric apparatus. In the determination of the specific gravity" of a liquid, the weights in air will suffice, but if the density 12 is desired, the weights "in vacuo" must be used.

Concerning gases, whenever it is necessary to weigh these, as for instance in the Dumas' method13 for the determination of molecular weight, the weights "in vacuo" are alone permissible because of the enormous error that would be introduced by employing the weights in air.

10 The volume of air displaced by the brass weights is equal to the sum of the weights divided by the density of the weights, i.e., 998 ÷ 8.0 = 124.8 c.c.

11 Specific gravity is defined as the ratio of the weight of a given volume of substance to the weight of the same volume of water, the temperature of each at the time of weighing being given.

12 Density is the mass of substance per unit volume, and in terms of the c.g.s., or absolute units, is the grams of substance per c.c.

13 For a brief account of Dumas' method, see Perkin & Kipping, Organic Chemistry. J. B. Lippincott Co., Philadelphia, 1918, 2nd ed., p. 38.

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