(adversely) by the hydroxy group. Increasing the ’chloride would displace increasing numbers of hydroxy groups and give increasing protection and increasing possibility of greater light emission. Kolthoff and Tomsicek (13), have pointed out that the ferricyanideferrocyanide potential may be affected by the presence of salts. They found that the effect was virtually independent of the type of anion. Thus, potassium nitrate and potassium chloride, and sodium nitrate and sodium chloride have identical effects on the potential a t the same ionic strength. The effects produced by cations, however, differ considerably. For alkali cations it decreases in the order Cs, Rb, K, Na. This is attributed to the degree of dissociation of the corresponding ferrocyanides. A larger effect was observed with divalent cations such as the alkaline earths. Inasmuch as the chloride and nitrate anions had distinctly different effects on
light emission a t the stoichiometric point, and sodium, potassium, and the alkaline earths produced no substantial difference, i t must be concluded that in determining cadmium, the lightemitting properties of the siloxene indicator far outweigh the possible changes in ferricyanide-ferrocyanide potential as a determining factor in the extent of light emission a t the stoichiometric point. The results included in Tables I to I11 indicate that this new and rapid method for the determination of cadmium permits the presence of ammonium, potassium, sodium, magnesium, strontium, calcium, barium, chromium, aluminum, and arsenate ions. The titration is best performed in a nitrate solution. Interference by chloride ion can be eliminated by standardization of the photometer in the presence of approximately the same chloride concentration as in the unknown solution.
LITERATURE CITED
(1) Ephraim, Fritz, “Inorganic Chemistry,” p. 835, Interscience, New York, 1947. (2) Kautsky, H., Kolloid 2. 102,9 (1943). ( 3 ) Kautsky, H., 2. Anorg. Chem. 117, 217 (1921). ( 4 ) Kautsky, H., Gaubatz, E., Ibid., 191, 388 (1930). (5) Kautsky, H., Herzberg, G., Ibid., 139, 142 (1924). (6) Kautsky, H., Hirsch, A., Ber. 64, 1614 (1931). (7) Kautsky, H., Thiele, H., 2. Anorg. Chem. 173, 118 (1928). (8) Kenny, F., Trans. N . Y . Acad. Sci. 16, 394 (1954). (9) Kenny, F., Kurtz, R. B., ANAL. CHEM.22, 693 (1950). (10) Kenny, F., Kurtz, R. B., Ibid., 25, 1550 (1953). (11) Kenny, F., Kurtz, R. B., Beck, I., Lukosevicius, I., Ibid., 29, 543 (1957). (12) Kolthoff, I. M., Furman, W. H.,
“Potentiometric Titrations,” 2nd ed.,
p. 318, Wiley, New York, 1931.
(13) Kolthoff, I. M., Tomsicek, W. J., J. Phys. Chem. 39, 953 (1935). RECEIVEDfor review July 16, 1963. Accepted November 26, 1963.
Analytical Applications of the Flame Spectra of the Rare Earth Elements and Scandium ARTHUR P. D’SILVA, RICHARD N. KNISELEY, VELMER A. FASSEL, RONALD H. CURRY,’ and ROBERT B. MYERS Institute for Atomic Research and Department of Chemistry, Iowa Stafe University, Ames, Iowa
b The analytical potentialities of the simple spectra emitted by the rare earth elements when ethanol solutions of their perchlorates are aspirated into fuel-rich, oxyacetylene flames are evaluated. The sensitivities of detection of the strongest lines are tabulated and data showing the application of these spectra to the analysis of complex rare earth mixtures are presented. The relative standard deviations for these analyses ranged from k l . 3 to 3.7% of the amount present.
T
relatively intense atomic line spectra emitted by the rare earth group of elements, which include scandium, in accordance with IUPAC recommended nomenclature (6), when ethanol solutions of their perchlorates are aspirated into fuel-rich, oxyacetylene flames were described in a recent communication ( 3 ) . The ultimate analytical utility of these spectra will be determined primarily by the detection sensitivities of the most sensitive lines and the degree of interference they enHE
Present address, Sperry Rand Research Center, Sudbury, Mass. 532
ANALYTICAL CHEMISTRY
counter. Both factors have now been evaluated, and the data are presented in this communication. One of the most appealing analytical applications of these spectra is the analysis of rare earth mixtures, since classical chemical methods cannot in general be used here. Although various emission and absorption spectrometric techniques have been successfully applied to this problem ( Z ) , it is apparent that serious deficiencies exist in their capabilities for providing accurate and complete analyses of all types of mixtures in a simple manner. The line spectra emitted in fuel-rich, oxyacetylene flames possess several distinctive advantages over other spectra heretofore used for the analysis of rare earth mixtures. First, the spectra are striking in their simplicity, compared to the arc or spark spectra. I t is therefore possible to achieve adequate spectral dispersion with small table-model spectrometers, whereas large, high dispersion spectrographs are required when arc or spark spectra are observed. Secondly, all of the rare earths except cerium exhibit line or band spectra in the flame of sufficient intensity to possess analytical utility. The group of rare earth elements which
evade detection by the optical absorptiometric technique ( 2 ) are readily observed in the flame spectra. Finally, the results reported in the present communication show that there is no evidence of interelement effects and that line interference, even in small monochromators, is rarely a serious problem. This contrasts sharply with the selective enhancement and absorption effects observed in x-ray fluorescent spectrometric measurements ( 2 ) . EXPERIMENTAL
Apparatus. The experimental facilities and operating techniques have been described ( 3 ) . For both the detection limit determinations and quantitative analyses, the scanning speed was reduced to 5 A. per minute. For this scanning speed, a chart speed of 1 inch per minute was used. The recorder response, photomultiplier tube voltage, and amplifier sensitivity were adjusted to optimal settings for the sensitivity determinations so that maximal signal to noise ratios were obtained. For the quantitative measurements a t higher concentrations, appropriate amplifier sensitivities and photomultiplier tube voltages were selected as discussed below. The sample aspiration rate for the
burner used in this tjtudy was 0.6 ml. per minute. The stoichiometric flame employed for the exc tation of the La0 emission was obtained by altering the acetylene and oxygen flow rates to 2.2 and 3.1 liters per min ite. Solutions. The rare earth perchlorate solutions in ethanol were prepared by dissolving 500 mg. of the oxides in 10 ml. of 1 to 1 perchloric acid (72%) and water. The resulting solutions were evaporated until a slurry of crystals was obtained. T h e solid mass nhich formed on cooling was dissolved in absolute ethanol (without he:iting) and diluted t o 100 ml. These stock solutions were diluted with absolute ethanol to yield the standrird solutions employed in the sensitivity determinations.
Limits of Detection and Interference Assessment. Table I shows the sensitivity of detection of either the most intense or the second ranking line of each element in the group. Additional sensitive lines which should find many analyticd applications are also listed. The sensitivities are expressed as the concentrations (parts per million by weight of the oxide in solutions prepared as discussed above) which yielded line intensities equal to twice the standard deviation of the background fluctuations (1). The strip chart recordings usec for these measurements were obtained a t solution concentrations approxiinately five times greater than the detsction limits. I n assessing interferences from other rare earths for the lines listed in Table I, consideration was given to all interfering lines of significant intensity within 2 A. of the line in question. Under the experiinental conditions employed, two lines separated by this wavelength interval could be completely resolved, even under rather unfavorable relath e intensity conditions. Several factors must be considered when the data in 'Table I are used to assess the feasibility of a given analysis. First, detection and determination sensitivities are dependent on instrumental and operating variables. Secondly, the relative intensity scale used in Table I is compressed at the low intensity end, because background corrections were not applied in the visual estimations (3). Thus, interferences from lines with relative intensities ess than 5 will in general not be as serious as the intensities would suggest. A third factor to take into account is the likelihood of encountering the interfering element in a typ cal sample. The Sc 4054.55 A. line interference with Ho 4053.92 A. line will seldom be encountered, since it is most unlikely that the truly rare element scandium will be found with holmium in sufficient amounts to cause an appreciable error.
Table I.
Wavelengths of Sensitive Lines, Detection Limits, and Interferences Sensitive lines Interferences -. -~ Wavelength, Relative Detection Wavelength, Relative Element A. intensitya limit, p.p.m. Element A. intensitya La 4187.32 40 4 4186.81 60 DY Er 4185.72 3 Tm 4187.62 50 Pr 4951.36 15 La 25 4949.77 3 Nd 1 4950.28 Sd 4950.72 1 Xd 4952.51 3 Nd 4924.53 5 Pr 4924.59 35 15 4924.06 Sm 10 4954.78 None 15 Sm Xone 4470.89 30 None 4783.10 5 50 Sone 4841.70 35 10 None 4883.77 40 Eu 4594.02 0.005 Nd 4594.67 1 100 Nd 2 4627.12 4626.50 90 Nd 3 4627.99 4661.88 Pr 1 4660.91 85 4663. 55 Sm 10 4058.23 Gd Er 4059.51 1 6 Tb 4060.38 1 4401.85 Sm 4401.17 25 20 10 4403.12 Sm 40 4403.27 2 Ho Tb 10 4318.85 10 Sm 4319.53 15 Sm 4326.48 4325.15 2 13 4326.13 2 Sm 4325.57 5 Gd 4327.10 4 Gd 4338.45 Sm 4339.35 5 8 2 Sm 4339.93 4211.72 100 0.5 Yone DY 4053.92 Ho Gd 6 70 4053.65 Lu 2 4054.45 Sm 1 4054.51 sc 4054.55 25 Gd 3 4054.73 4103.84 Y 4102.38 75 0.5 35 4103.88 6 DY 2 4104.88 La Tb 1 4105.38 Tm 4 105.84 65 4163.03 Pr 4163.01 2 60 sc 4165.18 3 3973.04 Er Eu 35 3971.99 10 3973.88 1 DY Sm 3974.66 10 2 4007.97 La 2 4007.66 60 Ho 20 1 4087.64 4087.35 La 4089.61 1 4151.11 Gd 45 1 4150.61 2 Sm 4151.21 2 4152.35 sc Tm Sone 3887.35 40 4092.90 0.5 Er 1 4094.18 60 4103.84 Ho 4105.84 65 75 4103.88 6 DY 2 La 4104.88 Tb 4105.38 1 Yb 3464.37 Tm 35 3462.20 1 Er 3987.99 0.1 5 100 3987.66 Sm 8 3990.00 Lu 3281.74 None 10 6 3312.11 None 15 3359.56 Sone 25 3376.50 None 13 Xone sc 3269.90 25 3273.62 None 12 3906.32 3907.48 Er 1 75 3907.11 Eu 8 3907.91 1 Tb 3911.81 3911 .80 Ho 2 75 Er 4020.40 60 4020.52 8 Er 4023.69 60 3 4021.96 Gd 5 4023.35 2 4023.72 DY (Continued) VOL. 36, NO. 3, MARCH 1964
0
533
~~
generally led to poorer precision, weaker lines or lower concentration solutions were utilized for determinations at the Sensitive lines Interferences higher concentrations. The summary Wavelength, Relative Detection Wavelength, Relative Element A. intensity" limit, p.p.m. Element A. intensitya in Table I1 shows the various combinations of lines and solution conY 3592.91 10 None centrations employed. Since the 4077.37 35 DY 4077.97 3 Er 4077.97 3 lanthanum lines possessed inadequate Gd 4078.71 6 sensitivity, the prominent L a 0 band La 4079.18 2 system with band head at 4418 A. was 4102.38 35 3 Ho 4101.09 5 utilized instead. A stoichiometric flame Ho 4103.84 75 Dv 4103.88 6 was used for this determination. 4128.30 30 H;, 4127, is 15 The standards employed for calibra4129.13 2 tion purposes approximated in total Eu Dy 4129.62 15 composition those normally encountered Eu 4129.74 15 4142.84 25 4141.52 2 in separative processes (Table 111). 4143.53 2 The analytical curves in Figure 2 are typical of those obtained from these a Relative intensities taken from Fassel, Curry, and Kniseley (3). mixtures. KOevidence of interelement effects was detected, even in mixtures containing up to a dozen other rare merits (pr, Nd, Gd, Tb, and y ) possess earths a t various concentration levels. moderately strong lines, but these lines Several instances of line interferences Table II. Experimental Conditions for were revealed in this study. -1s preAnalytical Calibrations suffer a higher degree of possible line interferences. cerium does not emit dicted by the interferences listed in Solution concn., Table I, the most sensitive line of lines of significant intensity and the % total gadolinium, 4401.85 -4.)is subject to poor sensitivity of the strongest line of Concn. rare lanthanum will limit its usefulness. serious interference from Sm 4401 -17 Analysis range, earth The prominent ~~0 band systems, A. The next most sensitive line of Element line, A. yo oxide oxide gadolinium, 4058.23 is free of emitted with greater intensity in stoiLa" La0 band 10-100 0.1 ,,biometric flames than in the fuel rich significant interference, but its intensity at 4423.1 1-10 1.0 environment, in spectral regions allows determination down to only 10%. Pr* 4951.36 10-100 0.1 2-25 0.1 relatively free of interference and thus The Tb 4326.48 line can be used for Nd* 4954.78 10-100 0.1 determining terbium down to I%, in the are more useful than the La 4187.32 A. 2-25 0.1 absence of samarium or gadolinium. line (4, 5, 7-14). Sm 4470.89 10-100 0.1 1-10 0.1 ~ ~ of complex d ~ R~~ Earth ~ i The Eu ~ 4594 02 A. line is very sensitive ~ i ~i~~~ possessing ~ ~ ~ and strongly ~ susceptible ~ to, self-absorpEu 4594.02 10-100 0.01 1-10 0.10 tion. Europium determinations a t the interference were selected Gd 4058.23 10-100 1.0 from the list of sensitive lines shown higher concentrations are best made on Tbc 4326.48 10-100 0.1 O.O1% Of the 1-10 1.0 in Table I and from the tables published Dv 4221.10 10-100 0.1 earlier (31, Consideration was also Procedure for Quantitative Calibra4211.72 1-10 0.1 tion and Analysis of Samples. As given to line interferences from nonHo 4120.20 10-100 0.1 rare-earth elements RThich may Occur shown in Table 11, the determination 4163.03 1-10 0.1 in significant concentrations in typical Er 4087.64 10-100 0.1 of individual rare earths in the con4007.97 1-10 0.1 centration range from 1 to 100% rerare earth mixtures. At the higher conTm 3887.35 10-100 0.1 centrations many of the most sensitive quires solution concentrations ranging 4094.18 1-10 0.1 lines were significantly self-absorbed. from 1% down to 0.01% total rare Yb 3464.37 10-100 0.1 3987.99 1-10 0.1 This effect is shown for the Y 3592.9 A. earth oxides. The standard operatLU 3312.11 10-100 0.1 ing procedure therefore involves t h e line in Figure 1. Since self-absorption Table 1.
(Continued)
%
-1.j
p i
S_ C .
Y
3269.90 .. .. 3911.81 3592.91 4077.37
1-10 10-100 1-10 10-100 1-10
1.0 0.1 0.1 0.1 0.1
Flame operated under stoichiometric conditions. *Scan for Pr and Nd done simultaneously. c TbaO, cannot be determined in range 1-10% in presence of Srn~Os. Q
Table 111.
Oxide La203 CeOz
50
PrsO11
Ndz08 S~ZOS Ell208
On the other hand, the Sm 4401.17 A. interference with Gd 4401.85 A. is serious, since these elements are usually found together. The line spectra of these elements differ in their potential analytical utility. Nine members of this group (Sm, Eu, Dy, Ho, Er, Tm, Yb, Lu, and Sc) possess strong lines with little or no serious interference. Five of the ele-
534
ANALYTICAL CHEMISTRY
Rare Earth Standard Mixtures
GdnOa
Wt. % 20 50 25 5
10 20 50 25 5
Y208
Tbr0.i DYzOs
10 50 25 5
2 5 2 10 50 25 5 1
HOrOa ErzOa
TmzOs YbnOs LUzos ScnOa
50
Total
100
5 10 5 2 10 50 25 5
10 50 25 5
10 50 25 5 10
2 2 2 10 50 25 5 5
1 2
lo0
100
1 6
2
2 10 50 25 5 1 2 5
100 lod 100 100
10 2 50 10 25 50 2 10 50 10 5 50 1 1 0 lod 5 0 100
2 10 2 1 1 1 1 1 1 1 2 2
25 25 25
100
1
I
2
5
10
25
100
50
/ -Ioo
901
Y
Pr 4951 36 A
U
0
25
75
50 Y203
100
CONCENTRATION (WT. %)
OXIDE CONCENTRATION (WT%)
Figure 1. Effect of concentration on self-absorption ofY 35‘32.9 A
preparation of 1% solutions, which may then be diluted to 0.1 or 0.01% solutions. The 1% solutions are prepared by dissolving 500 mg. of the sample in 10 ml. of 1 to 1 perchloric acid (72%) and water, using the procedure discussed under Solutions. If the approximate composition of the sample is not known, the wavelength regions covering the analytical lines are scanned to determine the applicable concentration range. These scans are performed a t relatively high amplifier sensitivities to assure detection of all of the constituents. Appropriate dilutions with absolute ethanol of the 1% uolution are then made for the determination of those elements requiring less concentrated solutions. The highest concentration standard solution within the concentration range to be examined is then aspirated into the flame. The amplifier sensitivity and photomultiplier voltage are adjusted so t h a t i,he analytical line for this solution produces a full-scale deflection on the recorder. The average background signal adjacent to the analytical line is suppressed to approximately 5% of full-scale deflection. The gain and background zero suppression controls are then adjusted so that a background signal of approximately 5% and a net line signal of 90% fullscale deflection is rtchieved for the highest concentration standard in the concentration range under examination. This procedure establishes the span of the recording system. The spectra of the samples are then recorded, using base line corrections to determine the net signal. The average value of triplicate runs is obtained. Appropriate bracketing standards (see Table 111) are then run to establish the analytical curve for the series of samples analyzed. This procedure is then repeated for each element to be determ ned.
Figure 2.
Precision and Accuracy. Table I V shows the results obtained from single runs made on different days of four yttrium - terbium mixtures. These data show t h a t most determinations can be made with a relative standard deviation of &2%. The accuracy of the calibrations was verified by analyzing a series of synthetic blends of oxides of widely varying composition. Table V shows the results obtained from this study.
Table IV.
Typical analytical curves
ACKNOWLEDGMENT
The authors express their appreciation to Constance Butler for her assistance during part of this investigation.
LITERATURE CITED
(1) Alkemade, C. Th. J., “Contribution to the Development and Understanding of Flame Photometry,” Ph.D. thesis, University of Utrecht, Holland, 1954.
Precision Data for Y203-Tb407Determinations
1 2 3 4 5
83.5 86.7 83.8 86.0 85.0
12.5 13.2 13.0 12.8 12.0
20.3 19.2 20.0 18.5 19.8
67.5 66.0 68.6 68.0 67.8
86.0 86.2 85.7 83.5 85.0
12.8 13.0 12.5 12.8 12.6
15.0 16.0 16.0 15.8 15.5
83.0 78.5 79.8 81.0 81.8
Av. R.S.D.,
85.0
12.7
19.6
67.6
85.3
12.7
15.7
80.8
1.6
3.7
3.7
1.4
1.3
1.4
2.7
2.2
a
Relative standard deviation. Table V.
Rare earth
Analysis of Synthetic Unknowns
A B C D Present Found Present Found Present Found Present Found 4.0 8.0 16.4 13.0 20.0 3.0 13.2 1.6 9.6 7.2 4.0
4.4 8.5 16.3 13.2 20.8 2.8 12.8 1.7 9.8 7.0 4.0
20.4 3.6
...
1.6
...
21.0 3.8
...
1.4
...
29.2 1.6
29.8 1.7
1.2 6.4 36.0
1.2 6.3 39.0
...
2.0 18.8 4.8 2.8 40.0
1.9 19.5 4.5 2.5 37.0
1.6 18.4 20.0
...
... ...
6.4 9.2 8.8 3.2 4.0
6.4 9.3 8.3 3.2 4.2
28.0
26.8
32.0
31.5
...
...
...
...
~~~~~~
...
I
.
.
...
...
...
1.8 19.0 20.0
...
...
...
~
VOL. 36, NO. 3, MARCH 1964
535
(2) Fassel, V. A., ANAL,CHEM.32, No. 11, 19A (1960). (3) Fassel, V. A., Curry, R. H., Kniseley; .?., Spectrochim. Acta 18, 1127 (lY6X). (4) Goto, H., Ikeda, S., Sudo, E., iVippon Kugaku Zasshi 81, 80 (1960). ( 5 ) Ishida, M., J , Chem. sot. j a p a n (Pure Chem. Sect.) 76, 60 (1955). ( 6 ) International Union of Pure and Applied Chemistry, J . Am. Chem. SOC. 82, 5523 (1960).
E..
(7) Menis, O., Rains, T. C., Dean, J. A., ANAL.CHEM.31,187 (1959). (8) Menis, O., Rains, T. C., Dean, J. A,, Anal. Chim. Acta 19, 179 (1958). (9) Piccardi, G., Spectrochim. Acta 1, 533 (1941). (10)Pinta, M., J . Re&. Centre Natl. Rech. Sci. Labs. Bellevue (Paris) 21, 260 (11) Poluektov, N. s., Nikonova, M. P., Ukr. Khim. Zh. 25,217 (1959).
(12) Possidoni de Albinati, J. F., Anales Asoc. Quim. Arg. 43, 106 (1955). (13) Rains, T. C., House, H. P., Menis, O., Anal. Chim. Acta 22,315 (1960). (14) Tremmel, C. G., master’s degree thesis, Iowa State University, Ames, Iowa, 1958.
RECEIVEDfor review September 6, 1963. Accepted November 26, 1963. Work performed in the Ames LaboTatory of the U. S. Atomic Energy Commission.
Principles of Turbidimetric and Nephelometric Titrations E. J. MEEHAN and GRACE CHIUl School o f Chemistry, University o f Minnesota, Minneapolis, Minn.
b The general principles of turbidimetric and nephelometric titrations are presented. Illustrative theoretical titration curves are calculated from exact light scattering functions. The conditions of particle size and relative refractive index under which useful titrations are possible are described. The principles are illustrated by titration of loe4 to 10%4 solutions of bromide with silver, and the reverse. The titrations are reproducible to 3% and have a relative error within 3 to 10%.
A
OF turbidimetric titration procedures has been described, among which may be mentioned titration of halides ( I ) , sulfate (12, I @ , and various other ions (2, 3, 6). The relative error usually is between 2 and 10% for 10-2 to l O - * X solutions. The present paper describes the theoretical and practical possibilities of nephelometric and turbidimetric titrations. The literature on such titrations is entirely empirical in regard to the optical aspects. The dependence of scattering upon particle size, shape, and size distribution usually is not considered but frequently the relations As an have been misunderstood. example of the latter situation, a statement often is found in the analytical literature to the effect that the “turbidity may be expected to be proportional to the area of the suspended particles.” Such a situation cannot occur in a titration (v.i.). I n this paper titration curves have been calculated from exact scattering theory for various titration conditions, and some experimental results are given for titrations of dilute solutions of silver and of bromide.
HOST
CALCULATED TITRATION CURVES
The symbols 7 and &(e) refer, respectively, to turbidity and to intensity scattered a t the angle 0 from 536
ANALYTICAL CHEMISTRY
the direction of the incident beam, with electric vector perpendicular to the plane containing the incident beam and the direction of observation. For definitions and experimental determination of 7 and &(e) and of scattering coefficient K reference is made to a previous paper (9). For spherical particles 7 is equal to n d K , where n is the number of particles per milliliter, and K and i~(e) depend only upon m and a (4). Here it may be noted that in order for 7 to be proportional to area (cf. Introduction), K would have to be independent of r. This is not the case except for particles much larger than are formed in any turbidimetric or nephelometric titration, as may be seen \from any table of values of K-e.g., ref. (14). The two simplest limiting cases of a turbidimetric or nephelometric titration correspond to constant size or constant number of particles in suspension during titration, without growth or flocculation of particles in the periods between successive additions of titrant. If the particle size remained constant after the first addition, the number of particles would increase linearly with addition of titrant up to the equivalence point. For dilute suspensionsLe., of the concentrations usually encountered in such titrations--r and i,( d ) are proportional to the number of particles and i t is evident without calculation that a plot of 7 or &(e) us. amount of titrant added (the volume change being either negligible or corrected for) would consist of a straight line intersecting a horizontal line a t the equivalence point. Most workers in the field seem to believe that such a situation should be observed generally, but this is not the case. On the other extreme, if the number of particles remained constant after the first addition, the particle size necessarily would increase with further addition up to the equivalence point. I n this case the variations of T and &(e) during titration are not obvious at all, except in the special case of small particles discussed below, but must be calculated from appropriate exact light scattering functions. I n an actual titration, still
assuming no growth or flocculation between additions, the behavior would be between t,he two extremes, but in practically all cases would be much closer to the second extreme. The following 7 and &(e) curves are calculated on the second basis. Monodisperse suspensions are assumed; the effects of heterodispersity and of growth or flocculation between additions are considered later. I n several practical situations i t is possible t o exercise some control over both factors. The fraction of equivalent amount of titrant added is denoted as f , and the fraction added in the first addition, at which all the particles (no per milliliter) are formed, is fo. The particle radius at any values o f f > fo, up to f = 1, is r f , and is given by Equation 1: r/
rdf/f~)~’~ (1) Small Particles. For small (Ray; leigh) particles characterized by r/X 5 1/20, K varies as a4 [cf. Equation 3 of reference ( 9 ) ] , t h e relation for ordinary values of m being valid within a few per cent up to a a b o u t 0.5. Therefore at a given A, T is proportional t o re; also, &(e) is proportional to re and is independent of e. Both 7 and i,(S) increase as ( f / f ~ ) up ~ to f = 1, and the value at a n y f < 1 is f 2 times the value at the equivalence point. There is no other situation in which 7 or &(e) increase so rapidly with f up to f = 1. The small particle case, therefore, is the optimum one with respect to the precise determination of the equivalence point, provided that 7 and/or &(e) are large enough to be measured accurately. The magnitude of each depends upon m, a, and the concentration of suspension (grams/ ml.), I n the following example, illumination with A,,, = 436 mp is assumed; extension to any other wavelength may be made easily. Taking the maximum value for validity of the small particle expression for K as a = 0.5, it is calculated that a t f = 1, =
1 Present address, Chemistry Department, Cniversity of Indiana, Bloomington, Ind.
