reasonably precise and accurate determinations, particularly with amounts in excess of a milligram. In the series of spectrophotometric titrations, no trend is apparent and none was found by statistical methods. In any case, it would be good judgment to standardize the EDTA with amounts of strontium not too different from the amounts expected. Estimates of the blanks were made experimentally and by statistical evaluations. When NaCl was not present, no blank was found by either method. In the case of the visual titrations in the presence of NaC1, an indication of a blank was found experimentally but when applied, only negative errors resulted; statistically, no adequate basis for a blank was found and none was used. In the spectrophotometric titrations, a blank of 0.019 ml of EDTA was found experimentally; the statistical estimate was found from a plot of the mean error (when no blank was used) as ordinate against the amount taken in the sets. A horizontal line through the average ordinate gave a reasonable fit to the points and, as
mentioned above, no trend was discernible; the average ordinate corresponded to a blank of 0.023 ml of EDTA, confirming the experimental value and this was applied. The absence of a blank when ethanol was present and the need for a blank when ethanol was not present did not find an acceptable explanation. A comparison of the two series of visual titrations shows that the presence of NaCl affects both precision and accuracy adversely but not ruinously. The spectrophotometric method appears to offer no advantage over the visual procedure. A more complete understanding of the applicability of EDTA titrations in practical situations would be gained from a systematic study of the effects of salts on the stability constants of the metal-EDTA and the metal-indicator complexes. RECEIVED for review March 13, 1969. Accepted August 25, 1969. Research supported in part by the New York University Graduate School of Arts and Science Research Fund.
Density Distribution of Solid Particles by a Flotation-Refractive Index Method Gordon H. Fricke and Donald Rosenthal Department of Chemistry and Institute of Colloid and Surface Science, Clarkson College of Technology, Potsdam, N . Y . 13676 George Welford
Health and Safety Laboratory, U.S.Atomic Energy Commission, 376 Hudson Street, New York, N . Y . 10014 IT WOULD BE DESIRABLE to have a method for obtaining the density or density distribution on small amounts of solid materials with a minimum of liquid. In this paper a procedure is described based upon flotation ( I ) in which miscible liquids mixed in the proper proportions are used to determine the density of the material which remains suspended, or the density range of a portion of the material. The density of the liquid mixtures can be determined by refractive index measurement once a suitable calibration curve has been obtained. These refractive index measurements require only a few drops of the liquid mixture and are much more convenient than the more conventional pycnometer measurements. The procedure is illustrated in this paper by the determination of the density distribution of a small sample of glass beads. The flotation method can be used to separate particles of different densities and to reduce the spread of a density distribution. Also, it may be used to remove solid impurities from solid particles, if the particles and impurities differ in density. EXPERIMENTAL
For the purpose of testing the method, glass beads (100-200 mesh, 149-74 microns obtained from Cataphote Corp., Jackson, Miss.) were used. The density of the glass beads was determined by the normal pycnometer method to be 2.4075 g/ml (relative error at 90% confidence level was 0.16 parts per thousand for three determinations using weights of solid materials ranging from 5.9 grams to 11.6 grams in a 25-ml pycnometer). Two liquids, dibromomethane ( d =
2.48, n = 1.538) and carbon tetrachloride ( d = 1.59, n = 1.463), were selected, because they had a large difference in refractive index, covered a reasonable density range, and were completely miscible. The densities were measured on mixtures of these two liquids using a pycnometer. The refractive index of these mixtures was obtained using a Bausch and Lomb Abbe-3L Refractometer. The density distributions of two samples of glass beads were determined by initially placing the samples (3.6995 grams and 5.0000 grams) into 25 X 150 mm test tubes with screw caps. A 30-ml portion of the more dense liquid, dibromomethane, was placed in the tube with the glass beads. T o aid the distribution of the glass beads between the bottom of the tube and the surface of the liquid, the tubes were centrifuged for approximately 10 minutes at about 2000 RPM (force at bottom of tube ~ 9 0 g). 0 The beads which had settled to the bottom of the tube were withdrawn with a pipet and placed in a tared weighing vial. The next desired density was obtained by adding carbon tetrachloride until the solution had the predetermined refractive index corresponding to the density. The tube was carefully swirled to prevent beads from adhering to the wall of the tube above the liquid. The procedure of adjustment of refractive index, centrifugation, recording of refractive index after centrifugation, and removal of the more dense beads, was repeated for each new density. Finally, the liquid, which had been placed in each weighing vial with the beads, was evaporated and the vials were reweighed to obtain the weight of beads within the narrow density regions. RESULTS AND DISCUSSION
(1) P. Hidnert and E. L. Peffer, National Bureau of Standards Circular 487, U.S. Government Printing Office, Washington, D. C., 1950. 1866
Refractive Indices and Densities of Liquid Mixtures. A number of empirice1 and semi-empirical equations are available by which the density, d, may be represented as a
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
Table I. Relationships between Refractive Index and Density Least Squares Fitted to d = a of Dibromomethane and Carbon Tetrachloride f(n) Name a b S.D.d X n Equation 1 - 14.1451 10.8095 2.25 2.25 -4.3704 21.8986 ClausiusMosotti (122 - l)(/?Z 2) Derivative of - 1.2911 1.8981 3.91 3 ClausiusMosotti ( n 2 - 1) Eykman -3.6908 8.7578 2.75 (n 0.4) (I? - 1) Gladstone-Dale -3.3355 IO. 8096 2.25 ( n 2 - 1) Laplace-2.3648 3.5485 3.08 Ramanathan
4- b . f(n) for Mixtures
+
+
(I
S.D.d
=
z.[d(observed)
-
S.D., X I O z b 8.63 3.50
S.D.b X I O z c 5.67 11.5
3.28
1.73
3.84
5.61
2.96 3.36
5.67 2.55
d(cslculated)12
Number of data points -2 S.D., = S.D.d 6 c S.D.* = S.D.d 6 where cll and cZ2are the diagonal elements of the inverse matrix obtained from the matrix formed from the normal equations. Refractive Indices and Densities are: 1.4964 (2.0289), 1.5045 (2.1180), 1.5158 (2.2418), 1.5298 (2.3919), 1.5310 (2.4072), 1 . 5 3 5 1 (2.4453), 1.5381 (2.4805).
function of the refractive index, n. (2). These equations are intended to be used with gases or liquid mixtures over narrow ranges of refractive indices and densities. The experimental data, for seven mixtures of dibromomethane and carbon tetrachloride, were least squares fitted to the straight lines for each of these functions as indicated in Table I. The first equation in Table I, a modification of the Gladstone-Dale equation, is used in this paper to calculate the density from the refractive index of the liquid mixture:
di
= -14.1451
+ 10.809sni
(1)
The error in the density, due to the conversion of refractive index to density by Equation 1, is seen from Table I to be approximately one part per thousand. The measurements which have been made are the weights of glass beads within narrow density regions, Aw,. The cumulative weight fraction is
0600
-
CI 0400
-
0200
-
0000
-
I
2 000
2 100
2 200
,
!
2 300
2400
-
d,
Figure 1. Cumulative weight fraction, Ci, cs. density, di (grams/ml) for two samples of 100-200 mesh glass beads
where the numerator is the cumulative weight of the particles having a density less than or equal to diand the denominator is the total weight of the sample. If the cumulative weight fraction, C t , is plotted LV. density, dr, this is the cumulative density distribution (3, 4). Because it is assumed that the function Ci is continuous, in the limit as Aw, approaches zero (3) where w iis the cumulative weight of particles having a density less than or equal to di. Figure 1 presents a plot of the (2) M. Kerker, “The Scattering of Light and Other Electromagnetic Radiation,” Academic Press, Inc., New York, N. Y., 1969, Chapter 9. (3) R. D. Cadle, “Particle Size: Theory and Industrial Applications,’’ Reinhold Publishing Corp., New York, N. Y., 1965, Chapter 1. (4) H. H. G. Jellinek, “Degradation of Vinyl Polymers,” Academic Press, Inc., New York, N. Y., 1955, Chapter 1 .
Experimental points were obtained by the flotation-refractive index method 0 5.0000-gram sample X 3.6995-gram sample - Least squares fitted beta distribution
cumulative density distribution. The points which are plotted represent the cumulative weight fraction calculated from the experimental data presented in Table I1 using Equation 2. The curves which are drawn are continuous cumulative functions fitted to the data. Another means of presenting the density data is as a frequency distribution (3, 4). The frequency distribution is a plot of frequency, F i , DS. density, di. The frequency, Fi,a t a density, di,is the slope of the continuous cumulative distribution a t dt-i.e., (4)
Ft can be evaluated from the cumulative distribution by graphical, analytical, or numerical procedures.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
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~~~~~~~~
~~
~~~~~
Table 11. Density Distribution Data for 3.6995-Gram Sample of 100-200 Mesh Glass Beads by Flotation-Refractive Index Method Experi- Cumula- Cumulamental tive tive Density, weight, weight, weight Calculated frequency, di,agrams/ Awn: W,, fraction, i ml grams grams Ci Fi,c ml/gram 1 1.9697 O.oo00 O.oo00 O.oo00 O.oo00 2 2.0249 0.0116 0.0116 0.0031 2 . 7 X lo-’ 3 2.0800 0.0124 0.0240 0.0065 0.0002 4 2.1556 0.0323 0.0563 0.0152 0.0161 5 2.2183 0.0267 0.0830 0.0224 0.1894 0.0412 0,7882 0.0695 0.1525 6 2.2670 7 2.2961 0.0555 0.2080 0.0562 1.5803 8 2.3556 0.6037 0.8117 0.2194 4.6999 9 2.3880 0.7791 1,5908 0.4300 6.9042 10 2.4205 0,9256 2.5164 0.6802 8.1380 11 2.4583 0.9198 3.4363 0,9288 4.9787 12 2.4615 0.1850 3.6212 0.9788 4.3439 13 2.4799 0.0783 3.6995 1.oo00 O.oo00 14 2.5030d O.oo00 3.6995 1.oooO O.oo00 Calculated from refractive index measurements using Equation 1. Experimental weight is the weight in the density interval di-1 to di. Obtained from Equation 10, see Figure 2. d Upper limit of density, beyond which there are no more dense glass beads, obtained by the flotation methods ( 1 ) using dibromomethane (d = 2.48) and diiodomethane (d = 3.33). -~ ~
Table 111. Density Distribution of 100-200 Mesh Glass Beads by the Flotation-Refractive Index Method Fitted to the Beta Distribution (Equation 5) 3,6995 5.oooo Sample weight. W , grams 1.5369 2.0745 Calculated volume V , (Equation 13), ml 2.4071 2.4102 Mean (average) density, d , grams/ml 0.17 1.12 Relative error (assume pycnometer d = 2.4075 grams/ml), ppt 10.389 9.238 Shape parameter, p 2.233 1.897 Shape parameter, q 2.4207 2.4298 Mode of distribution,” grams/ml 2.4068 2.4094 Mean density (- 5 of p and q), grams/ml 2.4073 2.4108 Mean density (+5 of p and q ) , grams/ml 2.4065 2.4082 Mean density (- 10% of p and q), gramsiml 2.4074 2.4113 Mean density (+10 of p and q ) , gramsiml a Mode (peak) of the noncumulative beta distribution =
z
x)
4,
Figure 2. Frequency, F, (ml/gram), GS. density, d, (grams/ml) for two samples of 100-200 mesh glass beads. Curves obtained by fitting cumulative data obtained by the flotation-refractive index method with a beta distribution (as in Figure 1) and using these least squares parameters to calculate F, (Equation 10)
This seemed appropriate because of the power of this distribution in fitting a wide variety of skewed shapes (7). The cumulative weight fraction can be expressed as the ratio of the incomplete beta function (numerator) to the complete beta function (denominator) :
li
f(z; P, 4 ) dz
C(ca1C)t
=
s,
wheref(z; p , q) is defined by Equation 6. f(z;p, q )
=
-
(6). This is combined with Equation 7 to obtain the + 4 - 2) mode in terms of density.
(P
A variety of procedures were considered for obtaining the continuous distributions from the data of Table 11. A plot of the experimental cumulative weight fraction data-i.e., C ics. d, as in Figure 1-revealed that the cumulative distribution appeared well behaved. Rather than trying to fit portions of the data with a polynomial as in interpolation procedures, all of the cumulative weight fraction data was fitted to an appropriate function of d,. The frequency distribution was obtained by differentiating the cumulative function (Equation 4). The density distribution was extremely skewed and the beta distribution (5,6) was used to fit the experimental data. (5) M. Abramowitz and I. A. Stegum, Eds., “Handbook of Math-
ematical Functions with Formulas, Graphs, and Mathematical Tables,” National Bureau of Standards, Applied Math. Series, No. 55, U. S. Government Printing Office, Washington, D. C., 1964. (6) G. J. Hahn and S. S. Shapiro, “Statistical Models in Engineering,” John Wiley and Sons, Inc., New York, N. Y . ,1967. 1868
zp-1 (1
- z)4--1
(6)
and
(7)
z
(’
(5)
f(z; P , 4 ) dz
where z is a dimensionless parameter having a value between zero and one, d,i, is the density below which there are no particles and d,,, is the upper limit of the highest density category which contained particles. Values of the shape parameters, p and q , were selected to give a least squares best fit to the experimental cumulative weight fraction data-ie., the sum of the squares of the deviations SSDC., q) was minimized, where
The complete beta function is evaluated as a quotient of gamma functions (5-9) : (9) The incomplete beta function is evaluated by the procedure of Osborn and Madey (8) or by use of a standard IBM program (9). (7) G. H. Fricke, D. Rosenthal, and G. Welford, unpublished work,
Clarkson College of Technology, Potsdam, N. Y.,1969. (8) D. Osborn and R. Madey, Math. Computation, 22, 159 (1968). (9) “System/360 Scientific Subroutine Package” (360A-CM-03X) Version 111, 4th ed., International Business Machines Corp., White Plains, N. Y.,1968.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
The ordinate of the frequency distribution is obtained by substitution of Equations 5 and 7 into 4:
The beta distribution least squares fit to the cumulative distribution is given in Figure 1. The frequency distribution represented by Equation 10 is given in Figure 2. As a means of checking the procedure used in fitting the density distribution, the average density, 6, is calculated. Only the total volume, V , of the particles must be calculated from the frequency distribution since the total weight, W , was obtained experimentally:
V
=
L"""du
It is assumed that an infinitesimal interval change in the volume is given as :
dw du = d The right hand side of Equation 12 may be obtained by equating the right hand sides of Equation 10 and Equation 4. Then the relationship in Equation 7 is used. The final result is written 1 zP-1(1 - z)B-l %nsX dz du = z(dmax - d r a i n ) dmin
s,
"s,
+
(13)
This equation was solved with the aid of the 16 point Gaussian quadrature formula ( 5 , 9 ) . To check the values of the shape parameters, p and q, as obtained from least squares fitting of the experimental cumulative density distribution to the cumulative beta distribution, the values of p and q were changed by & 1%, i5 %, and =klOx, The average density was recalculated with the new values of p and q. With a change of + 1 % in p and q the calculated density remained unchanged. All calculations were performed on an IBM S/360, Model 44 computer. Double precision (16 significant figures) was used and was necessary for the evaluation of the beta function, the least squares procedure, and the evaluation of the integral in Equation 13. The pertinent data for two independent determinations on the same lot of glass beads are given in Table 111. From the results summarized in Table 111, it is clear that the least squares cumulative and frequency distributions, when used to calculate a mean density of the glass beads, give results in good agreement with the pycnometer results. The error in the calculated mean densities is comparable to that expected when densities are determined by Equation 1. Also, examination of Figure 1 indicates rather good agreement between the calculated and experimental distributions. Comparison of the results for the two independent determinations indicates good agreement between these two sets of measurements. RECEIVED for review July 15, 1969. Accepted August 21, 1969. The U. S. Atomic Energy Commission, Health and Safety Laboratory provided a fellowship for one of the authors (G.H.F.) and partially supported the research. This work is taken in part from the Ph.D. research of Gordon H. Fricke.
Vapor Pressure Determination by Differential Thermal Analysis Herbert R. Kemmel and Saul I. Kreps Department of Chemical Engineering and Chemistry, Newark ColIege of Engineering, 323 High Street, Newark, N . J. 07102
DIFFERENTIAL THERMAL ANALYSIS (DTA) equipment for the determination of normal boiling points has been described by Vassalo and Harden ( I ) , Chiu (2)) and Kemme (3). The extension of these techniques to determine the temperaturevapor pressure function for pure liquids has been described by Krawetz and Tovrog (4, and Barrall, Porter, and Johnson (5). The latter workers reported measurement of vapor pressure between 30 and 760 Torr, and claimed an accuracy of 1 0 . 2 "C. The use of commercially available DTA equipment involves several modifications in hardware and special attention to operating techniques if the fullest capabilities of the method in time and material economy and precision are to be realized. Present address, American Cyanamid Company, Bound Brook, N. J. (1) D. A. Vassalo and J. C. Harden, ANAL.CHEM., 35, 132 (1962). (2) J. Chiu, ibid., 27, 1102 (1955). (3) H. R. Kernme, M. S. Thesis, Newark College of Engineering, 1963. (4) A. A. Krawetz and T. Tovrog, Rec. Sci. Instrum., 35, 1465 f 1962). ( 5 f - E - M . Barrall, E. S. Porter, and J. Johnson, ANAL.CHEM., 37, 1053 (1965).
DIFFERENTIAL THERMAL ANALYSIS EQUIPMENT
The DuPont 900 DTA instrument was the basic equipment used. This instrument has been previously described ( I ) . Thermocouple Circuits. The thermocouple circuits originally supplied use gold-plated brass feed-through plugs to connect the thermocouples to circuitry. Spurious potentials amounting to 1 0 . 4 mV are caused by temperature variations in the vicinity of the plugs. They were eliminated by installing a separate Chromel-Alumel, shielded extension line to the ice reference junction, connected through a ChromelAlumel plug. The use of a single bath for the ice point reference for both the furnace control and temperature measuring circuits produced interactions amounting to hO.02 mV. Separate ice baths eliminated this source of error. Potentiometer and Recording System Modifications. The manufacturer supplies an X-Y recorder with maximum sensitivity to the sample temperature signal from ChromelAlumel thermocouples of 0.4 mV per inch, equivalent to 0.01 inch per 0.1 "C. Determination of sample temperature to within 0.1 "C was impossible without modification. To achieve this discrimination, a Leeds & Northrup K-3 potentiometer and 2430-D galvanometer was connected in parallel with the recorder circuit (Figure 1). The X-Y
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