LITERATURE CITED
“Textbook of Quantitative Analysis,” 742. Macmillan. New York. 1948. r6j Mac’Nevin, W . ’ M., Urone, P. K., ANAL.CHEM.25, 1760 (1953). (7) Martin, A. J., in “Organic Analysis,” J. Mitchell, Mitchell, E. J. Proskauer, I. M. Kolthoff, A.’ A. Weissberger, eds., Chap. 1, pp. 2-64, Interscience, New York, 1960. (8) Ricciuti, C., Coleman, J. E., Willits, C. 0.. O., ANAL.CHEM.27.405 27,405 (1955). (9) Satterfield, C. N., ‘Bonnel, Bonnel, A. H., D.
(1) Denisov, E. T., Emanuel, N. M., Uspekhi Khim. 4, 365 (1958). (2) Egerton, A. C., Smith, F. L., Ubbe(London) lohde, A. R., Trans. Rov. SOC. ~ 2 3 4 ,433 ‘ (i935). (3) Furmanek, C., Manikowski, K., Roczniki Pahstwowego Zakladu Hig. 4,447 (1953) ; C. A . 48,8559 (1954). (4) Kolthoff, I. M., Meehan, E. J., Bruckenstein, S., Minato, H., Microchem. J. 4, 33 (1960). ( 5 ) Kolthoff,
I. M., Sandell, E. B.,
Ibzd., 27, 1174(1955). (10) Sauer, R. W., Weed, A. F., Headington, C. E., Preprint, Division of Petro-
leum Chemistry, ACS, 3, No. 3, 95 (1958). (11) Strohecker. R.. Vaubol. R. V.. ‘ Tenner, A., Fette ‘und Seifin 44, 246 (1937). C. A . 32,816 (1938). (12) Waish, A. D., Trans. Faraday SOC. 42,271 (1946).
RECEIVEDfor review April 26, 1961. Accepted June 29, 1961. Conference on Oxidation of Hydrocarbons in the Liquid Phase a t Low Temperature, U. S. Bureau of Mines, Bartlesville, Okla., May 23, 1961.
A New Falling Velocity Method of Density Determination for Small Solid Samples A. S. ROY Bell Telephone Laboratories, Inc., Murray Hill, N. J.
b A new method is described for determining the density of a solid sample of arbitrary shape and small size b y measuring its fall time in two viscous liquids of different densities, measuring the fall time of a sample of known density, and calculating the unknown density from two equations. This method is not restricted to measuring densities of solids with densities less than liquids available for flotation. The method has been investigated experimentally and found to measure densities of materials from approximately 2.5 to 10.5 with good agreement with reported densities.
T
HE SIQNIFICANCE of density measurements in solid state studies has been demonstrated recently by Smakula (9) and Horn (6). Frequently only small samples are available for investigation (3, 10) and density determination becomes difficult, as when dealing with single crystals, whiskers, isotopes, sintering, or hot pressing. Classical methods of measuring density have some limitations with respect to small samples. Weighing the sample and determining its volume, either directly, by measuring its dimensions, or indirectly, by substitution in fluid, auffers from loss of accuracy as the sample becomes smaller because of constant errors that do not decrease with sample size. The flotation method in which the sample is floated in a liquid of matching density, though suitable for small samples, is not applicable to samples having a density greater than 4 to 5 grams per ml., because no suitable liquids are available as yet. The falling velocity method, in which the gravitational velocity of a body in a viscous liquid is measured, has the advantage that the velocity of a small body
1426
ANALYTICAL CHEMISTRY
can be determined as accurately as that of a large one. However, the body must have a specific shape-.g., a sphereand i t is necessary to measure its dimensions, which again render the method inaccurate for small samples. These difficulties are removed by the method described below. PRINCIPLE
OF METHOD
Barr (1) has shown that the limitations of Stokes’ law for determining the viscosity of a liquid can be practically eliminated when two falling velocity experiments are conducted in two different viscous liquids on the same spherical body under the same geometrical conditions of the fall tubes. Stokes’ law states: v = 2/9
(’eu)
a sphere in Equation 1)for an irregularly shaped particle does not need to be known since i t cancels. This cancellation stems from the fact that in laminar flow any arbitrarily-shaped body has a certain shape factor which is a function of its geometry and orientation alone (1, 6, 7 ) . However, in order for Equation 2 to hold, the arbitrary body has to fall with the same orientation in the two experiments. Except for bodies with 3 axes of symmetry, bodies with arbitrary shapes tend to fall with one preferred stable orientation (4, 6) ; therefore, the condition for extending Equation 2 to arbitrary shapes is generally realizable. Thus, a generalized Stokes’ law equation can be formulated for a body of arbitrary (finite) shape falling along its stable orientation:
que
where u, p, and u are, respectively, the velocity, density, and the radius of the falling sphere; and u are the viscosity and the density of the liquid; and g is the acceleration due to gravity (all in cgs units). I n another liquid of viscosity p’ and density u’ a similar relation holds. Hence ( 1 )
in which t is the fall time along a certain path in one liquid and t’ that in the other liquid. The wall-effect and endeffect factors required to correct Equation 1 for applying Stokes’ law to finite media thus cancel out, since they are a function of only the geometrical features of the system (the sphere and the tube) in the case of extreme laminar flow ( I ) . It is significant to note that Equation 2 is independent of the shape of the particle. The shape factor (2/9 u2 for
in which K is a constant characteristic of the geometry and orientation of the body only, and one can obtain Equation 2 for a body of arbitrary shape from Equation 3 applied twice in two different liquids. If two separate pairs of such experiments are performed with two bodies, i and j, two equations of the type of Equation 2 are obtained, one with p,, t,, and t l and the other with p i , t,, and ti in which the subscript denotes the body in question. Dividing one of these equations by the other eliminates the viscosity ratio and results in
where
1
4
3
2
( 1 1 ( 1 1 1 1 1 / 1 1 CM
Figure 2. 1.
Quartz
Shape and 2.
Figure 1, Borosilicate glass fall tube
Consequently, function of p i , ratio 6:
pi
can be expressed as a and ut and the time
U,
where
Hence, by selecting a body j of known density pj as a standard and by measuring the liquid densities u and u t , it is possible t o evaluate the unknown density pi of body i, by making four measurements of fall time between two pairs of arbitrary marks on the two tubes. This constitutes the basis of a new method for measuring density of small bodies irrespective of their particular shape and density. EXPECTED ACCURACY
Liquids of high viscosity can be selected in order t o obtain extreme laminar flow as required for the accurate validity of Equation 2, and for obtaining large time readings. The method requires thermostating and provisions to eliminate air bubbles and convection currents. These precautions are also required in the flotation and the falling drop methods where accuracies of 1 part in 100,000 have been reported (a, 6). All these methods
relate to similar physical phenomena and employ similar procedures. They all measure relative quantities, and this permits the attainment of high accuracy when adequate care is exercised. Hence, it may be expected that the accuracy of the present method is limited in a similar way to that of the flotation and the falling drop methods. There is, however, one special requirement on the new method which is reflected by the denominator of Equation 6. By Equations 4 and 7 that denominator can be shown to be equal to ( u ~ - u ) / ( ~ ~ - u ' ) ;therefore, the difference ui-u in the densities of the two liquids should be as large as practicable. This requirement can be easily fulfilled as it is possible to choose from many synthetic liquids which are uniform, stable, and Newtonian, and are available in a variety of densities (from about 0.85 to 2.5 grams per ml. or more) and viscosities. I n exchange for this requirement, however, the method is effective for measuring densities of bodies which are too dense to yield to the flotation method and too nonspherical in shape to yield to the simple falling velocity method. EXPERIMENTAL
A simple experiment was carried out for demonstrating the feasibility of the method.
falling orientation of samples
Alumlnum
3.
Germanium
4.
Silver
Apparatus. Two fall tubes, about 3 inches in diameter and 20 inches high, were made of graduated borosilicate glass cylinders (Figure 1). The top of each cylinder was tapered into a female 12/18 ground joint inclined 30" from the axis of the cylinder; the male member of the joint was bent 120" and sealed to serve as a sample holder. Test specimens preloaded into this holder could be delivered into the center of the cylinder by rotating the holder a t the joint. A soft rubber hose extension served as a flexible liquid reservoir for the totally-enclosed liquid-filled cylinder. Materials. Pure ( a t least 99.99%) single crystals of germanium, quartz, polycrystalline aluminum, and silver shot were used for specimen materials. T h e specimens, listed in Table I and described in Figure 2, were rolled on fine sand paper and boiled in water to eliminate fragile edges and dust. For liquid media Indopol Polybutene H-25 (Amoco Chemicals) having a density of 0.8685 gram per ml. (u) was taken as the light liquid, and Fluolube T-80 (Hooker Chemicals) having a density of 1,9670 grams per ml.. ( u t ) , as the heavy liquid, both havlng viscosities around 40 poises. The densities were determined in 50-ml. pycnometers a t the bath temperature with an agreement of 2 parts in 10,000 between duplicate measurements. There are a number of other commercially available liquids that might be considered. For instance, Halocarbons (Halocarbon Products), Kel-F oils (Minnesota Mining) , Chlorowaxes (Diamond Alkali), silicones (General Electric or Union Carbide), and Paraplexes (Rohm and Haas). Procedure. The specimens were cleaned with CClr, dried, immersed in a small amount of the viscous liquid chosen, degassed under vacuum, and transferred with some of the liquid into the previously filled cylinder. The cylinder was then sealed by the liquid-filled holder and manipulated to collect the samples in the sample holder and any air bubbles in the rubber hose extension. The entire assembly was then immersed for several hours in a glass-sided water bath controlled a t 25.0" i 0.03" C. and illuminated by reflected light from a desk type fluorescent lamp. Each sample was dropped separately and the time intervals in VOL. 33, NO. 10, SEPTEMBER 1961
1427
where Table 1.
Largest Weight, Dimens., Sample Mg. Mm. Quartz 33 4.0 Al 36 3.5 Ge 18 3.2 2.4
Ag
0.9
Summary of Results
hverage Fall Time, Sec. Heavy Lt. liq. liq. 88.04 82.86 82.01 226.27
Comptd. Time Ratio, B
182.40 164.71 86.59 203.31
which the specimen fell between predetermined graduation marks (the 1400and 600-ml. lines that encircle the cylinder) were measured by electric timers. Although the timers could be read to 0.01 second, the personal error in judging by eye the time of approach of the specimen to the mark and instantly pressing the timer switch by hand was estimated to be about 0.1 second This error could be made smaller by averaging several time readings of the same samples in repeated runs.
-
( 8)
from which the unknown density pi was calculated using the experimental time ratio 8. The computed densities in Table I are close t o those reported in the literature. Considering the simplicity of the experimental procedure used, this agreement is satisfactory and can be accounted for by the self-correcting features of the method. Because both the standard and the unknown samples are dropped under similar conditions some proportional errors from timing, liquid nonuniformities, etc., cancel in part. The shape and wall factors (1) for these experiments would require a correction to Stokes’ law of a magnitude greater than 10%. However, even by the simplified procedure given here these factors were eliminated and a n accuracy of 0.1% was obtained. For higher accuracy refined experimental techniques as used in related methods are required. Better control of uniformity and temperature of the liquid (6),instrumentized fall time measurement-Le., by a photographic ar-
1428
ANALYTICAL CHEMISTRY
2.648(9) 2.698 (9) 5.3267 (9) 10.49(8)
Standard 10 46
Here, too, wide differences between the density values of the three standards are needed, and the variety of solids of densities ranging from 2 to 22 can be used to advantage for this purpose. Similarly, if falling velocities are measured in three rather than two liquids of measured densities, i t is not necessary to know the density of the standard body. I n this case two independent sets of equations of the type of Equations 6 and 7 can be obtained (one for the two liquids of densities u and u’, and the other for those of u and u‘), from which both p i and p , can then be calculated simultaneously. I n general, i t is necessary to know the density of only three materials for a complete solution of the system. Consequently, the employment of one additional standard body or liquid of known density can be used for cross checking the result.
DISCUSSION
The summary of the experimental data and the calculated results are presented in Table 1. Calculations were based on the germanium sample as a standard, the density of which ( p i ) was taken from the literature (9). On introducing the numerical values of the densities of the standard and the two liquids, Equation 6,becomes Pi
0.85106
2.652 2.700
rangement-and also the use of wider cylinders are indicated directions for improving accuracy. Specimens with a shape resembling that of a body with a pronounced single axis of symmetry, such as an arrowhead, mushroom, or section of a sphere, turned into their stable falling orientation faster than others. This is a n important factor for obtaining high accuracy by this method, and the selection or preparation of such shapes (even approximately) should be beneficial.
RESULTS
2.61018 - 0.8685 = 1 1.326%
1.9622 1.8827
Density, G./MI. Comptd Lit.
-
The method described for determining the density of small dense samples has several advantages. Since the method does not depend on the direct measurement of the volume or dimensions of the specimen, it does not involve a n error that increases as the specimen size is decreased, such as that resulting from pycnometric measurement. In fact, with smaller samples, laminar conditions improve, and this increases the accuracy with which Equation 3 holds. It is not necessary to know the viscosity of the liquids, the shape or wall effect ,factors, and calibrated floats of particular densities are not needed. The method lends itself to numerous For instance, when modifications. three rather than one solid standard specimens of known densities plJ pz, and p a are employed, there is no need to measure the densities of the liquids. These densities can be solved simultaneously from two independent equations of the type of Equation 4, one for standards Nos. 1 and 2 and the other for 2 and 3, yielding
(P3
and
-
d a r
ACKNO W LEDGMENI
Editorial comments from J. H. Scaff and F. J. Schnettler and some conversations with J. R. Fisher and R. H. Hansen are appreciated.
LITERATURE CITED
(1) Barr, G., “A Monograph of Viscometry,” Chap. VIII, Oxford Univ. Press, 1931. (2) Bauer, N., in Weissberger, A,, ed., “Technique of 0rg:mic ChemiRtry,” Vol. I, Part 1, pp. 171-3, Interacience, New York, 1959. (3) Decker, B. F., Kasper, J. S., Acta Cn st. 10, 332 (1957). (4) dans, R., Sitz A k . Wiss. Muchen 41, 191 (1911); Ann. D . Phys. 86, 654 (1928); Davies, C. N., in “Symposium on Particle Size Analysis,” Inst. Chem. Engrs. and SOC.Chem. Ind , London, Feb. 28, 1947. (5) Heiss, J. F., Coul, J., Chem. Eng. Prog. 48, 132 (1952). (6) Horn, F. H., Phys. Rev. 97, 1521 (1955). (7) Lamb, H., “Hydrodvnamirs,” 6th ed.: DD. 597-605, Dover Publication, New York, 1945. . (8) Silverman, A,, Insley, H., Morey, G., Rossini, F. D., Bull. National Research Council, No. 118, University of Pittsburgh, 1949. (9) Smakula, A., Sib, V., Phys. Rev. 99, 1744 (1959). (10; Walker, 1,. R., J . Appl. Phys. 29,318 (1958).
RECEIVED for review Mav 15, 1961. Accepted July 3, 1961. knnual ACS Meeting in Miniat,iire of the North Jersey Section at Seton Hall University, South Orange, N. J., January 30, 1961.
