Humboldt State University®Department of Chemistry

*From:*Smith, George McPhail.*A Course of Instruction in QUANTITATIVE CHEMICAL ANALYSIS for Beginning Students*. The MacMillan Company, New York (1921) pp 7-21.© Copyright 1998 R. Paselk *INTRODUCTION*7

I. WEIGHING* **Standards of Mass.**The fundamental standard of mass adopted by the United States is a cylinder of platinum-iridium kept at the International Bureau of Weights and Measures near Paris. Two authentic copies of this standard, of the same form and composition, are kept in a vault at the National Bureau of Standards, and they are used only when needed to verify the secondary standards of the Bureau.- Mass standards are usually called "weights," but it should be realized that, while the weight of a body depends upon its attraction by the earth, the mass of a body is a property inherent in the body itself. Since, however, the masses of bodies are proportional to their weights at the same locality, we may compare the masses by making a comparison of the respective forces of gravity on the bodies under consideration. The purpose of weighing, then, is to compare the quantity of matter in a specific object with the quantity of matter in a given standard - a gram or kilogram weight. The comparison is made on the balance, by means of the lever principle, by suspending the object at one end of a beam, and the weights at the other end of the beam, the beam being virtually a kind of lever.
**The Balance.**The beam of the balance is supported on a central knife-edge, usually of agate, which rests upon a plane agate plate; and two pans for supporting the masses to be compared are vertically suspended from stirrups, each of which has an agate bearing which rests on a knife-edge fixed at one extremity of the beam. The arms of the balance are so graduated that a rider (of known weight) can be placed on the beam at any required distance from the central knife-edge.- If the three knife-edges are allowed to press continually upon their agate bearings, they soon become blunted, and wear furrows in the bearings. In order to prolong the life of the knife-
- * For more detailed information on
the subject of weights and weighing, the student is referred
to
*Circular of the Bureau of Standards, No. 3*(3d edition, 1918), from which much of the following is taken. 8 *QUANTITATIVE CHEMICAL ANALYSIS*- edges and bearings, the balance is provided with a "release" which separates the knife-edges from their bearings when the balance is not in use. If the balance shows signs of stiffness in the motion of beam and pans, the fault should be investigated at once. The defect may be due to an accumulation of dust between the knife-edges and their bearings; to the blunting of the knife-edges; or to the wearing of furrows in the bearings. To prevent the accumulation of dust, and also to prevent the interference of air currents while weighing, the balance is inclosed in a glass case.
- In order to render small movements of the beam perceptible, there extends downwards from its center a long pointer which multiplies the rotational displacement. When equilibrium is established the lower end of the pointer should come to rest in front of the zero of a scale which is located immediately behind this end.
- The conditions which must be satisfied by a good balance are: (1) The balance must be consistent. It must give the same result in successive weighings of the same body. This condition depends upon the trueness of the knife-edges. (2) The balance must be accurate. At rest the beam must be horizontal when the pans are empty, and when equal weights are placed upon the pans. This condition depends upon the equality of the two arms. (3) The balance must be stable. The beam after being displaced from its horizontal position must return to its horizontal position. This condition depends upon the adjustment of the center of gravity. (4) The balance must be sensitive. It must show even a very small inequality in the two masses on the scale pans. This condition depends largely upon the length of the arms. (5) The balance must oscillate with reasonable rapidity. Short beams oscillate more rapidly than long ones.
- The analytical balance will perform excellent service under the proper conditions, but great care in its use is essential if its accuracy is to be relied upon. It should be located in a room
*INTRODUCTION*9

- that is free from dust and fumes, and should stand upon a support that is free from shocks and vibrations.
**The Use and Care of the Analytical Balance.**The following rules embody the main points to be observed in the use and care of a balance.- (1) Each student must feel a personal responsibility for the proper use of his balance; the carelessness of any one may render inaccurate the work of all who use the same balance.
- (2) Before use, with clean pans, the adjustment
of the balance should be
*tested*by the student. - The balance is properly adjusted only if the following conditions are fulfilled: (a) The spirit level or plumb bob inside the balance case should show that the balance is level; (b) the mechanism for raising and lowering the beam should work smoothly; (c) the pan arrests should just touch the pans when the beam is lowered; (d) the pointer should rest at zero when the beam is raised; and also when it is lowered, with the pans supported; and (e) the pointer should swing equal distances on either side of the zero when the beam is set in motion without any load on the pans. In the latter case, if the variation does not exceed two divisions on the scale, it is sufficient to allow for the small zero error, without adjustment.
- (3) The beginner should not attempt to make adjustments himself, but should apply to the instructor in charge.
- (4) In order to set the beam in motion, it
should first be lowered so that the pans rest upon the pan arrests;
the latter are then cautiously lowered, and, in case the beam
fails to swing, it can be set in motion by means of the rider.
There is, however, a " trick " in lowering the
*pan supports*so that the oscillations of the pointer will have the required amplitude.* - The pans should be arrested and the beam raised before any change is made in the load or weights on the pans except in the
- * It is not necessary to employ very
long swings; an amplitude of two scale divisions is probably
sufficient, even in very exact work. See in this connection Horace
L. Wells,
*Jour. Amer. Chem. Soc.*, vol. 42, p. 411 (1920). 10 *QUANTITATIVE CHEMICAL ANALYSIS*- case of the small fractional weights, when it is only necessary to arrest the pans. The object to be weighed and the heavy weights should be placed in the middle of their respective pans, since a heavy load near the edge of a pan is apt to cause trouble some oscillations.
- The beam and stirrups should never be left upon their knife-edges, and the motion of the beam should be arrested only by means of the pan arrests, and only when the pointer is passing the center of the scale; otherwise the knife-edges and their agate bearings are subjected to an unnecessary strain.
- (5) The weights should be cared for as well as the balance. They should be handled carefully, and only with the forceps provided; they should never be touched with the fingers. In weighing, the weights should always be placed upon the same pan, and they should be taken in the order in which they occur in the box, the larger ones first; and the weight of the object should be recorded by noting the vacant spaces in the box. The record so obtained should be checked as the weights are removed from the pan. In this way errors are not likely to occur.
- (6) Analytical samples should not be placed
directly upon the balance pan, and the object to be weighed should
not be warmer or colder than the air in the balance case. Currents
of hot air will tend to buoy up one arm of the balance, and also
to cause that arm to expand in length.* If the object is colder
than the atmosphere of the balance case, moisture may condense
on its surface. If the body to be weighed is likely to be electrified
(
*e.g.*a glass weighing tube), it should be allowed to stand for some time after it has been wiped, before weighing. - (7) The balance case should be closed while weighing with the rider, so as to avoid currents of air.
- As soon as the object is apparently balanced by the weights the beam should be raised and again lowered into place, and the
- * For instance, a platinum crucible which appeared to weigh 20.649 g. when
- warm, weighed 20.6920 g. when cold.
*INTRODUCTION*11

- observation repeated. This insures the proper alignment of the beam and pans.
- (8) In using weighing bottles or tubes, the vessel should be weighed together with its contents. The sample should then be removed without loss, and the vessel and contents again weighed. The difference in weight indicates the quantity taken. The weight of a tube, recorded at some previous time, should always be confirmed before weighing out a new sample from it.
- (9) Errors in weighing should fall well within
the limits of the experimental error due to the analytical operations.
If, for example, an error of 0.001 g. were made in weighing out
a gram sample of fireclay containing 0.25% Of MgO, the resulting
error in the determination of the magnesia could be no greater
than 0.1% of its value, which is negligibly small. A 1 mg. error,
however, made in weighing the 0.0069 g. Of Mg
_{2}P_{2}0_{7}would involve an error of over 14% in the magnesia value, and this would be intolerable. - (10) Finally, if anything at all appears to be the matter with a balance, the instructor's attention should at once be called to the fact.
**Determination of the Rest-point.**The beam and stirrups are first lowered upon their knife-edges by slowly turning to the left the milled head at the front of the balance case; then the pans are released by gently pressing inwards the small button, also at the front of the case; and, with the beam swinging smoothly, a consecutive record is made of the number of scale divisions traversed by the pointer on either side of the center. The swings to the left are recorded as negative numbers and those to the right as positive numbers; in the determination of the rest-point, one more reading must be made on one side than on the other, and all of the readings must be consecutive. Upon dividing by 2 the algebraic sum of the averages for the two sides, the quotient is the rest-point of the balance for the case under consideration,*i.e.*the position on the scale at which the pointer would finally come to rest.12 *QUANTITATIVE CHEMICAL ANALYSIS**Example:*Left Right -4.8 -4.6 +2.7 -4.4 +2.5 Average: -4.6 Average: +2.6 Rest-point = -1.0. - Two methods of procedure are now open to the operator. He may either make his weighings with reference to this observed rest-point or he may adjust the balance so that the observed rest-point is the actual zero of the scale. The first method is preferable, unless the rest-point is more in error than one scale division. The rest-point is apt to change, and it must be determined each day, or even more often.
**Methods of Weighing.**Weighings smaller than 0.005 9. (or 0.01 g.) are made with the rider. When the arms are divided into five divisions, a 5-milligram rider is used; in general, the rider should weigh as many milligrams as there are large divisions between the central knife-edge and the right-hand stirrup support. Each large division on the beam then corresponds to a milligram.*Ordinary Method.*The object to be weighed is placed upon the left-hand pan of the balance and weights upon the righthand pan, until, finally, the further addition of 5 mg. (or 10 mg.) more than counterbalances the object. This weight is then removed, the balance case closed, and the rider adjusted so that the pointer swings equal distances on either side of the rest-point. This method of weighing is very common, and it is sufficiently accurate for ordinary analytical work. If necessary, the rest-point of the unloaded balance should be determined before each weighing.- In special cases, as in the calibration of a set of weights, it is important to make more accurate weighings. It is here best to
*INTRODUCTION*13

- employ the method of weighing by the use
of deflections. Though this method may appear to be laborious,
the labor is more apparent than real. The
*sensitiveness*of the balance may be found by adding a small weight to one of the pans and noting how far the rest-point is deflected from its former position. Then, instead of adjusting small weights until the rest-point is brought to the proper place, we merely note the deflection from this point and calculate from the sensitiveness the weight that would be needed to bring the rest-point to the desired position. The sensitiveness will, in general, be different at different loads, and, especially with a very sensitive balance, it must be redetermined from time to time. For very accurate work, it is advisable to determine the sensitiveness at each weighing. *Method of Weighing by the Use of Deflections.*(a) The restposition of the unloaded balance is determined, as already described. Let us suppose this to be at +0.1.- (b) The deflection of the rest-point per
milligram, or the sensitiveness of the loaded balance, is determined.
The object to be weighed is placed upon the left pan, the weights
on the right pan. When the weights have been adjusted so that
an additional 0.005 g. (or 0.01 g.) would be too much (
*e.g.*weights =11.216 g.), the balance case is closed, and the rider adjusted until the pointer swings on both sides of the zero of the scale. The rest-point is then found; for example, at +o.8. The rider is moved one milligram division, in this case to the right, and the rest-point again determined; at, say, -2.1. That is, the rest-point is deflected +o.8 - (- 2.I) = 2.9 divisions by 1 milligram, in the case of that particular load. - (c) The weight of the object is calculated. In this case, the object weighs 11.216+x g. The rest-point of this load is displaced 0.8-0.1=0.7 scale division. Since 2.9 scale divisions correspond to 1 milligram, 0.7 scale division win correspond 0.7/2.9 = 0.24 mg. Hence the weight of the body is 11.216+
14 *QUANTITATIVE CHEMICAL ANALYSIS*- 0.00024=11.21624 g. These calculations may be summarized in the formula
- in which z represents the rest-point of the unloaded balance, a, the rest-point with not quite enough weight on the right pan; and b, the rest-point with a milligram more on the right pan than corresponds to a.
- Analytical balances will rarely indicate with certainty less than 0.0001 g. Hence, although the weight may be calculated as above to the fifth decimal, it should generally be rounded off by dropping the fifth decimal and raising the fourth decimal one unit when the dropped figure exceeds 5.
- In certain cases, as in the calibration of volumetric measuring apparatus, it is necessary for the weight found to be independent of any inequality in length in the beam arms. In such cases, and in the determination of absolute weights (reduction to weights in vacuo), one of the following methods should be used.
*Method of Weighing by Transposition.*When the arms of a balance are nearly equal, the method of transposition furnishes a more accurate comparison than that of substitution (see below). It requires about the same amount of time for the observations, but the true weight is not shown in so direct a manner. Before and after the transposition, the rest-point of the loaded balance is noted and the added weight that would be needed to bring the rest-point to the desired position is determined as already described.- Having weighed the object first in one pan, then in the other, if we let W be the true weight, a the weights required to counterbalance the object when it is on the left pan, and b the weights required when the object is on the right pan, then, according to the principle of moments:
- That is,
- whence
*INTRODUCTION*15

- Therefore the true weight is the square root of the product of the two observed weights.
- Weighing by transposition is recommended for work of high precision in which it is also desirable to calculate the rest-point from several swings of the pointer. In other cases, substitution is generally to be preferred.
*Method of Weighing by Substitution.*Here the object, placed on the right-hand pan, is approximately balanced by a suitable tare (weights, wire, beaker containing shot, etc.) on the lefthand pan, and the rest-point determined. The object is then removed, and weights are added in its place until the rest-point is restored to its former position. These weights are necessarily the same in value as the object for which they substitute, irrespective of differences in the arms.**The Calibration of a Set of Weights.***Fairly accurate weights can readily be purchased, and for most analytical work the inaccuracies of the better class of weights are negligibly small in comparison with the errors of experiment, and the imperfections in the analytical processes.- An analyst, however, should know that his weights are sufficiently accurate, and for this reason he should calibrate the weights. The errors in weighing due to imperfections in the weights can easily be reduced to 0.0001 g. The weights should be tested at periodic intervals, say once or twice a year, depending upon the frequency with which they are used.
- In special cases,
*e.g.*in the calibration of volumetric apparatus, absolute weights may be required, but for general analytical work these are not necessary. If the weights are consistent with one another, their absolute values have no influence upon the accuracy of an analysis. - Duplicate and triplicate weights of a set should be so marked that they can be readily distinguished. A satisfactory method is to use one and two conspicuous dots on duplicates, and one,
- * In this connection, see
*Circular of the Bureau of Standards, No. 3,*and also - T. W. Richards,
*Jour. Amer. Chem. Soc.*, vol. 22, p. 144 (1900). 16 *QUANTITATIVE CHEMICAL ANALYSIS*- two, and three such dots on triplicates, - stamped by means of a small punch. The designations of the weights being tested are conveniently inclosed in parentheses, to prevent their confusion with the numerical data for use in the calculations. And it is customary to express the results in terms of corrections to be applied in the use of the weights, rather than in terms of the total values; the plus sign for a correction indicating that the mass of a weight is greater than its nominal value, the minus sign that it is less. In using the weights, these signs are then treated in the ordinary algebraic manner. For example, if (10') = 9.9987, (10") = 10.0008, and (5) = 5.0002, the respective corrections are -1.3, +0.8, and +0.2 mg.; the nominal value of the weights is 25.0000 g., but their corrected value is 25.0000+(-1.3+0.8+0.2) mg., or 24.9997 g.
- The weights are best calibrated according to their apparent masses as determined by comparison with brass standards in air; that is, in terms of "weight in air against brass." This method is advisable, owing to the prevailing use of small platinum and aluminum weights in connection with the larger weights of brass.
- In the calibration of analytical weights, the rest-point should always be calculated from several swings of the pointer, and the individual calculations should be extended to five decimals. If the beam arms do not differ in length by more than 0.001%, the simple method of deflections may be used; otherwise it is necessary to make use of transposition or substitution. The substitution method involves less work in calculating than that of transposition, but if it is used it is desirable to have a second set of weights to furnish the tares.
- In the case of an ordinary analytical set of weights, from 20 g. to 5 mg., the following comparisons should be made, with the use of the rider, as in ordinary weighings.
*INTRODUCTION*17

*A**B*THE SMALL WEIGHTS* THE LARGER WEIGHTS (500) against (20) against (10 ')+(10 ") (200) against (100 ')+(10 '') (10) against (10 ") (100 ') against (10 '') (10 ') against (100 ') against (5) against (50) against (2) against (1')+(1") (20) against (10 ')+(10 ") (1') against (1") (10 ') against (10 ") (1') against (1''') (10 ') against (5) against (Rider at 5) *C**D*One-gram piece, (1'), against

Ten-gram piece, (10 '), against an absolute (known) standard of the same nominal value.

- The weight of each of the smaller pieces is calculated from the data in terms of one of them, say the ten-milligram piece (10'), as a standard; and, in the same way, that of each larger piece in terms of one of them, as the ten-gram piece (10'). Then, in order to express all the weights of the set in terms of a common standard, the values of the fractional weights (in terms of their standard) are added, and the result compared with their total weight as found in terms of a one-gram piece, say (1').
- Suppose, for example, the values of the small
weights are: (500)=490.03, (200)=200.07, (100)'=100.10, (100")=100.00,
(50)=49.87, (20)=19.98, (10')=10.00, (10")=10.03, (5)= 5.00,
and (Rider at 5)=5.00 -
*i.e.*while, in terms of the one-gram piece (1'), their collective weight is 1.00l65. Then, in order to express the values in terms of (1'), each of them should be multiplied by (1.00165)/(0.99908) = 1.00257. If, however, in terms of the ten-gram piece (10'), (1')= 0.99954, the provisional - * The designations S(500), etc., are here used to represent collective weights of the nominal values indicated; in this instance, (200)+(100')+(100")+(50)+(20) +(10')+(10")+(5)+(Rider at 5).
18 *QUANTITATIVE CHEMICAL ANALYSIS*- value of each fractional piece may be multiplied directly by (1.00165 x 0.99954)/(0.99908) = 1.00211, to express their weights in terms of that of the ten-gram piece (10').
- What we actually do, assuming (10') is found to have an absolute value of 9.99970 g., is to multiply the individual values of the larger pieces, based upon (10') as a standard, by 0.99997, and those of the fractional weights, expressed in terms of the ten-milligram piece (10'), by [(1.00165 x 0.99954)/0.99908] x 0.99997 = 1.00208. In this way, we finally arrive at the individual values of the pieces, both large and small, in terms of the absolute standard.
**Errors Due to Inequalities in Length in the Beam Arms.**In the preceding discussion it has mainly been assumed that the two arms of the beam are equal in length. This is not really the case. It is mechanically impossible to insure perfect equality. To find the relative lengths of the arms, place (corrected) weights of the same nominal value - say, 50 grams - upon each pan, and bring the balance into equilibrium by means of the rider. Interchange the weights on the two pans, and again bring the balance into equilibrium by means of the rider. Call the two weights*W*and*w*, and let l and r respectively denote the additional weights required for equilibrium on the left and right sides. Then, on the first weighing,*w*+l=*W*; and, on the second weighing,*W*=*w*+*r*. Let*L*and*R*respectively denote the length of the left and right arm. Then from the law of levers,*L*(*w*+*l*)=*RW*: and*LW*=(*w*+*r*)- Solving each of these equations for
*W*, and equating the results, we find that - whence,
*INTRODUCTION*19

- Suppose, for example, with use of true weights, that the weighings were found to be:
Left Right (50) = (20)+(10')+(10'')+(10''')0.13 mg. (20)+(10')+(10'')+(10''') = (50)+0.19 mg. - Here, then,
*l*=-0.00013,*r*=+0.000019, and*w*=50 g. Consequently if in the above expression we let*R*= 1, we have *i.e.*L: R= 1.0000032:1- With this ratio L/R =1.000032 a weight
*w*on the left pan of the balance will be equivalent to a weight*w*x 1.0000032 on the right pan. Hence, if an object on the left pan balances the weight 50 g. on the right pan, the weight of the object is 50.0000 x 1.0000032 = 50.0016 g. - an error of only 0.003 per cent. There is therefore no need to apply the correction. Each balance has its own constant*L*:*R*for a given load; the numerical value of the ratio varies with the different loads. - Most analytical balances do not require a correction on account of inequalities in the arms; the lengths are usually sufficiently exact. Anyhow, if the weights are always placed, say, on the right pan, such a correction is unnecessary in ordinary analytical work, because the weights observed are proportional to the true weights, and the ratios obtained are not affected.
**Errors Due to Atmospheric Buoyancy.**Two bodies equal in weight at the same time and place contain the same mass or quantity of matter only if the weighing is carried out in a vacuum, or if the bodies have the same volume. The latter is hardly ever the case.- A body weighed in air is buoyed up by a pressure equivalent to the weight of the air which it displaces. Suppose that exactly 10 grams of platinum (sp. gr., 21.55) are weighed in air with
20 *QUANTITATIVE CHEMICAL ANALYSIS*- brass weights (sp. gr., 8.4). Then 0.45 cc.
of air, at say 20° and 760 mm.,
*i.e.*about 0.00054 g., are displaced by the platinum; while the weight of air displaced by 10 grams of brass is O.OO143 g. Hence, the weight of brass which will be required to counterpoise 10 grams of platinum is 1O+(O.OO143 -0.00054) = 1O.0009 9. - The arithmetic of the above calculation is summarized in the formula:
- in which
*w*represents the apparent weight of the object;*s*, the specific gravity of the object;*s*_{1}, the specific gravity of the weights; and , the weight of a cubic centimeter of air under the conditions prevailing at the time of the experiment. - To illustrate the effect of the buoyancy
of air on a few common substances, when weighed with brass weights,
it may be stated that the error per gram of substance weighed
is 0.1 mg. for ferric oxide (sp. gr. 5.12); 0.3 mg. for Mg
_{2}P_{2}0_{7}(sp. gr. 2.40); and 0.4 mg. for sodium chloride (sp. gr. 2.13). - In ordinary analytical operations we have
to deal with
*differences*in weight, and with*ratios*, not with absolute weights. When the amount of a precipitate is determined from the difference in the weight of an empty crucible and of the crucible plus the precipitate, the buoyancy correction is not needed for precipitates with a specific gravity near that of the substance undergoing analysis. If, however, the specific gravities are widely separated, it may be worth while to correct for buoyancy. For instance, since the specific gravities of iron ore and barium sulphate are nearly equal, it would be a waste of time to correct for buoyancy in determining sulphur in an ore of iron. On the other hand, in standardizing a solution of silver nitrate by precipitating silver chloride from a specific*weight*of the solution (use of weight burettes), the buoyancy of air may cause an error of 0.1%. **Summary.**The foregoing discussion clearly illustrates the advisability, in work of an accurate nature, of always estimating*INTRODUCTION*21

- the effect of the various sources of error on the final result. Any of these errors can then be neglected, provided it is sufficiently small in comparison with the error derived from other sources. The chief sources of error involved in weighing are those due to: (1) variations in the rest-point of the balance; (2) inconsistent weights; (3) inequalities in length in the beam arms; and (4) atmospheric buoyancy. In weighings making any pretense to accuracy, the following points should be noted:
- (1) The rest-point of the unloaded balance should be determined and made use of in each weighing.
- (2) The weights should be calibrated, and periodically checked for consistency.
- (3) The errors due to inequality in length in the beam arms may be neglected in ordinary analytical work.
- (4) Although the correction of weighings for buoyancy can almost always be neglected in general analytical work, owing to the much larger errors associated with the preparation of precipitates for weighing, it is advisable in weighing bulky glass apparatus (potash bulbs, etc.), to use a similar piece of apparatus as a counterpoise. This will serve to eliminate any errors due to variations in temperature, pressure, and humidity during the course of the determination.

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