concentration -34%) or up to at least 1.6 ml. (HC1O6 concentration -IS%), quantitative recoveries are obtained. The standard deviation for fluoride diffusions using N a F is =!=0.030 in the 0.5- to 3.0-fig. F-range for diffusions from 0.3-n11* The standard deviation for diffusions from 1.0-ml. sample volumes is *0.057. For fluoride in dental enamel, the standard deviation persists & *0.050 (121 determinations on 29 samples run a t
least in triplicate). This higher standard deviation may be due, a t least partially, to sample nonhomogeneity* LITERATURE CITED
( 1 ) Beckman Instrument Co., Instruction
Manual for Beckman Model B Spectrophotometer, Bulletin 2 9 1 4 , Fullerton, Calif. ( 2 ) Conway, E. J., “Microdiffusion Analysis and Volumetric Error,” 4th ed., pp. 26-32, Macmillan, New York, 1957.
(3) Frere, F. J., Microchem. J. 6, 167 (1962)* (4) Hall, R. J., Analyst 85, 560 (1960). (5) Rowley, R. J., Farrah, G. H., Am. Ind. Hyg. Assoc. J . 23, 314 (1962). (6) Singer, L., Armstrong, W. D., J. Dental Res. 41, 910 (1962). (7) Stegemann, H., Jung, G. F., 2.Physiol. Chem. 315,222 (1959). (8) Wharton, H. W., ANAL. CHEM.34, 1296 (1962).
H. WHITXEYWHARTON The Procter & Gamble Co. Miami Valley Laboratories Cincinnati 39, Ohio
A Modified Flotation Method of Density Determination for Small Solid Samples SIR: When it can be used, the flotation method provides the most reliable measurement of the density of a small solid sample. The sample is added to a suitable liquid solution, whose composition is then adjusted until the sample just remains suspended in the fluid. The density of this equilibrium liquid solution, which is identical with that of the solid, is then measured, usually by a pycnometric or hydrostatic method. The flotation technique cannot be applied directly t o solids with a density greater than about 4.3 grams per ml. because transparent liquids with densities above that figure are unavailable. Methods based upon direct measurement of volume or upon measurement of liquid displacement have been used for such solids, but relative errors less than 1Oyocan be achieved only by the techniques of ultramicroanalysis (1). Recently, Roy (6) proposed a method for determining the density of a small solid sample of arbitrary shape and high density from measurements of its velocity of fall, relative to that of a standard solid, in two viscous liquids of different densities. He used germanium as the standard material and demonstrated the feasibility of the method with measurements on specimens of silver, aluminum, and quartz ranging in size from to 10+ ml. On the basis of an unspecified number of replications on each of the three specimens he claimed a relative error of O.lyc.
Table I.
Sample Weight sample, M a , mg. Weight film, M I , mg. Density composite, pc, grams per ml. Density sample, pa, grams per ml. (Eq. 4) Av. pa, grams per mi. Literature pa, grams per ml.
scripts s, f, and c designate sample, film, and composite specimen, respectively: M , = M a AIf (1)
+
5= Pc
Figure 1. Composite specimen of solid sample and polymer sheet
This report describes a technique that retains the relative simplicity of the flotation method and has been successfully applied to five different samples covering a density range of 5 to 21 grams per ml. and a volume range of lod4to ml. Although it does not provide the accuracy claimed for the falling velocity method, it is still superior t o displacement methods for small samples. EXPERIMENTAL
Method. By mounting the small specimen of solid material through a slit cut in a sheet of polymer film of appropriate size, density, and chemical inertness (Figure l ) , one can prepare a composite specimen with any desired density between t h a t of the film and the sample. The density of this composite specimen can be determined by the flotation method and the density of the solid sample can then be calculated as shown below. Let the symbols N , V ,and p designate mass, volume, and density and the sub-
Densities at
pa =
+ Pa
(2)
Kf
(3)
Pf PfPc
(Pf
- Po)
+
(4) Pf
The values of M , and M, can be determined by direct weighing on a microbalance; p, can be determined by conventional methods with a large specimen of film; p . is the measured density of the composite specimen. Materials. The lower density component of the composite specimen was 4-mil Mylar film (E. I. du Pont de Nemours and Co.) with a density of 1.3920 grams per ml. Benzotrifluoride ( p 1.2, Hooker Chemical Corp.) and tribromofluoromethane ( p ,- 2.7, The Dow Chemical Co.) were used to prepare the equilibrium flotation solutions. Preliminary experiments showed that Mylar film was unaffected by immersion in these liquids for several days. Specimen materials were germanium (99.99+%), vanadium (Fisher Scientific Co., c.P., fused), nickel (99.5%), silver (99.95%), and platinum (99.96%). Procedure. A weighed piece of the metal specimen was inserted through a slit in a piece of Mylar sheet. Any sample shape was satisfactory t h a t could be retained mechanically by the polymer sheet, as show-n schematically
-
30” C. by Modified Flotation Method
Ge
V
Ni
4.793 4.793 2.954 2.954 2.561 2.562 5.31 5 . 3 2 5.32 5.32 (6)
4.949 4.310 3.215 2.683 2.628 2.654 6.21 6 . 0 9 6.15 6.11 ( 4 )
6.388 6.382 5.282 5.324 2.577 2.577 8 . 7 0 8.89 8.80 8.89 (2)
Ag 8.308 8.300 6.818 6.870 2.648 2.649 10.21 10.47 10.34 10.50 (3)
Pt 3.443 3.448 3.580 3.610 2.565 2.560 20.76 21.09 20.93 21.45 ( 6 ) ~~
VOL. 35, NO. 3, MARCH 1963
~
407
in Figure 1. Pieces of Mylar film were then trimmed from the edges of the sheet until the composite specimen just sank in a solution of benzotrifluoride and tribromofluoromethane whose density had been arbitrarily adjusted to approximately 2.5 grams per ml. The composite specimen was blotted dry and its components were weighed. After reassembly, the composite specimen was added t o a stoppered cylinder that contained about 15 ml. of the fluorocarbon solution. The cjlinder was mounted in a water bath maintained at 30” C., and the density of the solution was altered as necessary by addition of one or the other of its components. The densities of the solution and composite specimen were considered t o be identical when, after they were shaken, the sample remained suspended in the column of solution for 20 minutes, neither rising t o the top nor sinking to the bottom. The density of the equilibrium solution was measured with a calibrated 10-ml. pycnometer. RESULTS
The data are summarized in Table I, with values of the sample densities calculated by use of Equation 4 and values taken from the literature. Each member of the pairs of duplicates
represents a reassembly of the composite specimen, sometimes with a new piece of Mylar sheet, and a fresh equilibration procedure. DISCUSSION
The error in the unmodified flotation method depends only upon the sensitivity of the equilibration procedure and upon the error in the measurement of the density of the equilibration solution; it is independent of the size and shape of the solid sample, neither of which must be known. In the modified method described here, the density of the solid sample is not identical u-ith that of the equilibrium solution, but must be calculated from it through a function that involves other parameters (see Equation 4), errors in the determination of which are reflected in the error of the calculated value. I n the present work p i is fixed by the choice of Mylar film and p c is adjusted t o lie at a level slightly below the density of tribromofluoromethane. Under these conditions there is, in principle, neither a lotr-er limit to the sample size that can be used nor an upper limit t o the density that can be determined. However, as seen from Equation 4, the relative error in the calculated sample density should
increase both with a decrease in the size of sample for a solid of given density and with a n increase in density for a sample of given mass. However, even with 3.4 mg. of platinum, which is the most unfavorable example in Table I, the relative error \vas only 2.4%. ACKNOWLEDGMENT
The author acknom-ledges the skillful assistance of Xancy A. Parker in performing the measurements and calculations described in this paper. LITERATURE CITED
(1) Kirk! F’; L., “Quantitative Cltramicroanalysis, p. 299, Wiley, Xew York,
1950. (2) Kirk, R. E., Othmer, D. F., “Encyciopedia of Chemical Technology,” T’ol. 9, p. 271, Interscience, Sew York, 1952.
(3) I b t d . , T’ol. 12, p. 426. (4) Ibid., Vol. 14, p. 583. ( 5 ) Roy, A. S.,ANAL. CHEX 33, 1426
(mi).
( 6 ) Smi!jlells, C. J., “Metals Reference Book, Vol. 11, p. 636, Interscience, iVew York, 1955.
EDWARD L. SIMONS General Electric Research Laboratory Schenectady, S. Y.
T ha I1 ium- Colu mn Dissolved Oxygen Ana lysis: Recommended Values of TlOH Equivalent Conductance care n as tahcn to minimize interference SIR: alecurate values of the equiLby carbon dioxide, has yielded apprecialent conductance of thallous hyable improvement over the previous droxide in water are required if the data. However, these meaiurements metallic thallium reaction-column for cover a range of concentration (10-3 to dissolved oxygen analysis (a, 8) is to 10-*31) that is higher than normally enbe used under conditions Ivhere calcountered in dissolved oxygen microibration against another analytical proanalysis. *in accurate conductance cedurs of demonstrated accuracy is equation niuzt be used for cxtrapolanot possible. Such a situation may tion from the eqerimentally detcrniined occur, for emmple, a t T ery low oxygen range. The Fuoss-Onsager conducconcentrations, or in circunistances tance equations ( I , 2 ) >as modified for where the advantages peculiar to the ionic aswciation, are the most suitable thallium column make this the only analytical method that can be used. Even n hen compariyon with other *O / ’ methods is poisible, it is deqirable to make the thallium column analyzer independent of other procedure., since, in many cases, the thallium column will give the more accurate results if used properly. Previous determinations of thallous I hydrouide equivalent conductance by J 0.1 Ostwald ( 7 ) and Hlasko and Salitowna 0.005 0.01 0.1 1 10 100 Dissolved Oxygen Concentration (D.p.m.) ( 3 , 4)are incorrect, probably because of carbon dioxide difficulties. A recent Figure 1 . Equivalent conductance of thallseries of precision conductance measure- ous hydroxide at 25” C. for use with thallium ments on TlOH solutions ( 6 ) , in which dissolved oxygen analyzer
-
-
408
ANALYTICAL CHEMISTRY
for this purpose. Since the use of these equations involves a fair amount of computational labor, the calculated results are communicated for the convenience of persons concerned n ith analytical applications. Values of equivalent conductance a t 25” C. are given in Table I a t concentrations from to 10-2*1P, corresponding to dissolved oxygen concentrations from 0.008 t o 80 p.p.ni. These s e r e obtained from the FuossOnsager equations by using previoualy determined values of the parameters (6). K h e n numerical substitutions are made, the equations reducc to:
+
- 122.68 (CY)’ ‘ 123.69 C Y log(c-0 302.41 c y ] /
A = [273.00
+
(1
+ 3.0 cJi2),
(1)
lvhere logf=‘
=
-1.0182 [I
(CY)’
’/
+ 0.9558 (C-,)”*],
(2)
A is equivalent conductance at concentration c (moles per liter), and y is the degree of dissociation. At concentrations below 10-3JI, the degree
