Principles of Nephelometric Titrations and Application to Titration of Dilute Solutions of Bromide E. J. Meehan” and S. Yamaguchi’ Deparfment of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455
Calculations are presented of the variation of intensity of light scattered during a titration in which a nonabsorbing colloidal suspension is formed. The calculations are based on exact Mie theory for monodisperse homogeneous spheres of various refractive indices and illustrate the choice of optimum particle sire and optimum angle for measurement of scattering, as well as the effect of addition of preformed colloidal material before the titration. The principles are illustrated by the titration of M in mixtures of 2-propanol and bromide, as dilute as 5 X water with an average accuracy of 1-2%.
T h e term “nephelometric titration” as used in this paper means a titration in which the relative intensity of light scattered by the suspension which is formed in the titration is measured at a specific well defined angle from the direction of the monochromatic incident beam. The general principles have been presented earlier ( I ) . The present paper consists of two parts. T h e first part is entirely optical in nature. I t summarizes the results of extensive theoretical calculations based on exact Mie theory for scattering by homogeneous spheres. T h e calculations illustrate the choice of optimum scattering angle and of optimum particle size of the colloidal material, and show the effect of addition of “seed”-that is, preformed colloidal material of a given particle size-before t h e titration. Many of the calculations apply to a refractive index ratio (defined below) around 1.76, which corresponds to a suspension of silver bromide in water a t wavelength 436 nm, but illustrative calculations are summarized for other ratios. In the calculations, the suspensions are assumed to consist of monodisperse spheres which scatter independently. T h e effects of nonsphericity and polydispersity are referred t o later. T h e assumption of independent scattering is valid for t h e dilute suspensions which are used in practice. T h e second part of the paper presents results of titration of dilute solutions of bromide with silver nitrate. I t indicates to what extent the theoretically optimum situation may be approached in practice for this particular system. Calculated Titration Curves. T h e quantity il(0) is proportional to t h e intensity scattered per particle a t angle 0 from the direction of the incident beam, which is unpolarized and of unit intensity. The electric vector in the scattered beam is perpendicular to the plane containing the incident beam and the direction of observation. Vh (dimension cm-’) is given by Equation 1:
Nxzil(8) v,, = -
8n
where N is the number of particles per mL and X is wavelength in the medium surrounding the particles. Vs, is the vertically polarized component of the Rayleigh Ratio (see Ref. 3, p 38, p 503) at scattering angle 0. The concentration of suspended material, g-mL-’, is c. The density of the colloidal material is assumed to be the same as that of the bulk material in the calculation of c from N and radius r. The specific scattering Present address, Kao Soap Co. Ltd., Wakayama-shi, Japan. 2268
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
is Vou/c. For the method of calculation of i,(0) and Vouand their relation to the Mie scattering functions, see Ref. 2 or 3. In a nephelometric titration, it is sufficient to measure the relative, not absolute, intensity. However, knowledge of the absolute scattered intensity is useful because it allows comparison of the scattering by a suspension of given c with the scattering by water, or other suspending medium; the latter quantity determines the ultimate limit of detection of suspended material. The scattering by pure, dust-free water with the same state of polarization specified above for i,(S) is independent of 0. The quantity which for a pure liquid corresponds to Vw,u for a suspension is the Rayleigh Ratio Rw referred to above. For water R W is equal to 2.32 x cm-l and 8.65 x IO-‘ cm-’ a t wavelengths 436 and 546 nm, respectively. (It may be mentioned that the scattering of water is less than t h a t of many organic liquids, being that of benzene for example. In the Experimental section it is shown that the use of 2propanol as co-solvent is beneficial in the case of silver bromide. RgOfor this substance has not been measured but it may be expected to be roughly the same as that of methyl ethyl ketone, which is about twice that of water.) Rgo is compared later with V,, of various suspensions. T h e quantity a is 2rrIX; X is &Inrn,where A, is vacuum wavelength and n, is the refractive index of the medium. The refractive index ratio m is n/n,, where n is the refractive index of the colloidal material. T h e present paper is restricted to real values of n, corresponding to nonabsorbing substances. Because it is conventional to use a relatively concentrated solution of titrant, it is not necessary to correct for the change of volume during titration. (Such correction, of course, is trivial.) Silver bromide absorbs slightly at 436 nm. In earlier work (41, it has been shown that the value of h in the complex refractive index ratio m ( l - ik) a t 436 nm in small-particle suspensions is 3-5 X lo4, which is small enough that in respect to scattering the particles behave as nonabsorbing. The two limiting cases of these titrations correspond to (a) constant particle size and increasing number of particles and (b) constant number and increasing size, in both cases with no agglomeration or flocculation between additions of titrant before or after the equivalence point (EP). If case (a) existed, it is obvious without calculation that the titration curve (TC), that is, relative intensity vs. volume of titrant, would consist of a rising straight line intersecting a horizontal straight line a t EP, regardless of particle size. Actually the truth lies between (a) and (b) but is closer to (b). Therefore in the calculations it is assumed that when no seed is added, all the particles are formed upon addition of the first increment and these grow upon subsequent additions. I t is assumed that when seed is added, the number of particles remains equal to the number added as seed. Consider first the situation in which no seed is added and denote the fraction of equivalent amount of titrant added as f. The particle radius up to f = 1 is given by Equation 2,
Percent T i t i a t e d
Figure 2. Calculated titration curves, constant number of particles after addition of 1 'YO of titrant; m = 1.76
Relative Particle Size, a
Figure 1. Specific scattering vs.
cy
at various angles; m = 1.76
in which ro and f o correspond to the initial increment. A similar relation connects a f and ao. For small ("Rayleigh") particles with r I X/20, corresponding to cy 5 0.314,i1(8) is independent of R and proportional to r6:
(3) Therefore Vsu/c varies as f , and because c also varies as t , the scattered intensity a t any 0 increases as f' up to f = 1. This is the optimum situation in regard to shape of T C and location of E P , but the sensitivity is not as great as it is for somewhat larger sizes. For example, suspensions of silver bromide and barium sulfate in water a t A, = 436 nm have m values equal to 1.76 and 1.23, respectively. The respective densities are 6.47 and 4.50 g-mL-'. From Equations 1 and 3 the respective values of Vsu/c are found t o be 18.7 and 3.39 for cy = 0.314. Comparison of the former with for water shows that this silver bromide suspension scatters as strongly as water for c about lo-' g-mL-l. T h e corresponding c for barium sulfate is about 6 X lo-' g-mL-'. As a increases above the Rayleigh region, the scattering becomes concentrated in the forward direction and depends in a n increasingly complex way upon cy and m. Figure 1 illustrates the situation for m = 1.76 up to cy = 3.0 (Figures 1-6 are derived from computer-generated plots. The plotting increment of a in Figure 1 is 0.05.) For smaller or larger values of m, the curves are generally similar to those of Figure 1but are displaced to larger or smaller a , respectively; some representative TC's for other m are given later. Specifically for m = 1.76, a maximum specific scattering occurs a t a about 2.4 for angles close to zero, which angles are easily accessible experimentally with a laser source. Beyond a = 3, fluctuations become increasingly more pronounced. Evidently u p to a = 2.4, there is a great increase in sensitivity with increasing Q , especially a t the forward angles. For example a t 0 = 30", V,, is about 8000 at (Y = 2.4, about 400 times as great as the value a t cy = 0.314. However, if a at EP is too large, the T C has an oscillatory and practically useless character as is shown in Figure 2. (In this and Figures 3-6 the curves are calculated with titration increments of 2 % .) Figure 2 is calculated on the assumption that all the particles are formed on addition of 1 % of titrant; f o = 0.01 and cyo = O.l(O.1)l.O. In this case a1 = = 4.64. For cyo up to 0.5, the TC's have positive slopes up to EP and t h e location of E P is easy. A value of a. about 0.5, corresponding to cy1 about 2.3, is preferred for
maximum sensitivity. For larger CY" the location of EP is difficult or impossible. As mentioned earlier, these calculations apply to monodisperse spheres. If the system is heterodisperse, the fluctuations would be masked to some extent, the more so the greater the heterodispersity. Theoretical calculations can be made easily for any assumed distribution of sizes, but would not be of general value. The angular scattering pattern for nonspherical particles is generally similar to that of spheres, but a nonsphere of given volume in general has an angular pattern corresponding to a sphere of somewhat larger volume. Thus the same general conclusions drawn from Figure 2 apply to nonspheres. In the calculation of Figure 2 it is assumed that the value of cyo can be chosen by the experimenter. Actually this usually is not possible, because particle sizes in colloidal suspensions depend in general upon reactant concentrations, mode of mixing, and other factors which need not be listed here. Moreover the samples may contain constituents which affect particle size. For this reason it may be a great practical advantage to add seed material which has been prepared separately under such conditions as yield the desired size. Assuming as stated earlier that N remains equal to the value added as seed, the increase in cy during titration is given by
(4) where cy, and cyf refer to seed and suspension, respectively, and w is weight fraction added as seed with respect to the constituent being formed in the titration. At E P ,
+w (w) 1
ff1=
a,
(5)
Thus a l / a , = 4.7, 2.2, and 1.3 for w = 0.01, 0.1, and 1, respectively. Figures 3-5 give curves at. various 0 for these values of w. All the curves apply to m = 1.76 and cy, in each case is chosen such that a1 = 2.3 so that there are no oscillations a t the forward angles. In fact u p to 0 = 40°, the curves rise smoothly to EP. I t is of interest to note that for R L 50", there is a pronounced maximum before EP. This arises from the first maximum visible in Figure 2 which is not compensated by the second maximum as it is for 0 5 40". T h e total scattered intensity a t a given 0 increases with increase of w, but too large a value of w obviously is undesirable because i t would correspond to a small relative change in scattering. From Figures 3-5 it appears that values of w up to around 1would be appropriate with R 5 40'; the smallest 0 consistent with the instrumental optics should be chosen. Figure 6 gives calculated TC's for w = 0.01, a , = 1.0 and m = 1.3. Curves (not shown) for m = 1.5 and 1.2 are generally similar, being ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
2269
I
46,
4c
/?-
163I
c
-3
ZC
4C
6C
3c
:c
22
X
Percent Titrated
Figure 3. Calculated titration curves, constant number of particles after addition of 1 % seed of CY, = 0.493; m = 1.76 (scale X 10 for 8 I 70")
x c_
32-
W C c
nLee "0
I
2C
40
6C
EC
OC
2C
Percent Titrated
Flgure 4. A s Figure 3, but 10% seed of as = 1.04
% 7 50 10c
2c
Percent Titrated
Flgure 5. A s Figure 3, but 100% seed of
8 I 60')
CY,
= 1.83 (scale X 10 for
more and less structured respectively. For m = 1.3 and 1.2, satisfactory titrations are theoretically possible a t small 0, with a wide latitude in the choice of w and CY,.For the same L(: and CY, as in Figure 6, the TC's for m = 1.76 (not shown) are very complex with several maxima and minima before EP. All the above conclusions are entirely optical in character and are based on the assumptions of monodisperse spheres of negligible solubility in the surrounding medium, all of which being formed on addition of the first increment or added as seed, with no agglomeration or growth between additions of titrant. These are matters of colloid chemistry rather than optics and will be characteristic for each chemical system. The experimental part which follows considers the specific case of silver bromide. 2270
Percent Titrated Figure 6. As Figure 3, but 1 % seed of CY, = 1.00; m = 1.3 (scale
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
10 for 0 L 50")
Table I. Solubility of Silver Bromide in Solvents of Dielectric Constant D at 25 " C Solvent PKS, D 12.3 80.4 water 30% ethanola 13.2b 62.6 62% methanol 13.7b 50.4 50% ethanol 13.7b 50.4 48.5% acetone 13.7b 50.4 70% ethanol 14.4b 39.1 methanol , 15.2 32.6 90% ethanol 15.2b 29.0 82% acetone 15.2 29.0 ethanol 16.1 24.3 acetone 18.7' 20.7 a Weight percent (wiw). At 20 " C . At 23 'C.
Ref 5 6 6 6 6 6 7,8
6 6 5 9
Titration of Bromide with Silver. In titrations in which complexation reactions are unimportant, t h e solubility is greatest at EP. From the solubility of silver bromide in water (pK,, = 12.3 a t 25 "C), it is calculated that 0.7, 7, or 70% of silver bromide is dissolved a t E P when initial [Br-] is lo", or lo-' M, respectively (negligible volume change). Since the present interest is in methods applicable at great dilution, solvents in which the solubility is substantially smaller than in water must be used. For initial [Br-] = lo-' M, a pK,, I 18 corresponds to solubility at EP 5 1% . Table I lists pK,, in solvents of various dielectric constants D. While the solubility in 1-propanol, 2-propanol and tert-butanol apparently has not been measured, the values of D of these solvents (20.1, 18.3, and 10.9, respectively) indicate t h a t mixtures with water should be suitable for titrations of very dilute solutions. All the experiments reported in this paper were done in mixtures of water-organic solvent. The refractive index ratios in these mixtures range from 1.70 to 1.74 and thus the calculated scattering is very similar to that in water. Under most conditions the addition of silver nitrate to dilute aqueous solutions of bromide yields suspensions in which the particle size increases rapidly for several minutes (IO). Such systems obviously are useless for titrations and i t was necessary to establish conditions such that a practically constant size and scattering intensity would be reached within some seconds after addition of titrant. Such suspensions are called "stable" for the present purpose. EXPERIMENTAL Reagents. Analytical reagent grade potassium bromide and silver nitrate dried at 110 "C for 2 h were used to prepare 0.1 M
stock solutions in distilled water. The silver nitrate solution, standardized against sodium chloride and stored in the dark, was used to standardize the potassium bromide solution. All organic solvents were analytical grade except isopropanol which was obtained by distilling the technical grade. No substances reacting with silver nitrate were present in any of the organic solvents.
Table 11. Particle Growth in Titration in Presence of Seed, rs = 46 n m % seed r,,nm(calcd) r j , nm (obsd)
"^
,, J'J --
0
1
10
100
...
213 51
103 53
58
40
57
2cc
Percent Titrated 1
Figure 7. Experimental curves in 80% 2-propano1, [Br-] = M, I t = 30 s, 0 = 25'. Amount of seed: (1) none, (2) 1%, (3) l o % ,
e
v
RESULTS AND DISCUSSION Stability studies at lo4 M bromide with less and more than the equivalent amount of bromide gave the following results. Unstable suspensions are formed in acetone (50%) in the absence and presence of any of the protective colloids listed in Experimental, methanol (5070) in the absence and presence of PVP, and ethanol (5070,70%). Suspensions stable within 30 s or less are formed in methanol (€io%),ethanol ( 8 0 % , go%), 1-propanol ( 8 0 % ) ,and 2-propanol (80%,90%); in the latter group, the addition of protective colloids is unnecessary. Effect of A m o u n t of Seed. Figure 7 gives the results of titrations in 2-propanol (8070,It = 30, [Br-] = M) with 0, 1, 10, and 100% seed. The end points are located more easily in the presence than absence of seed because of the increased slope before EP. In these four and most other titrations, there is a small positive slope after EP, which undoubtedly is due to a slight agglomeration upon increase of concentration of silver ion. The T C with 1% seed, cy, = 0.9, is drastically different from the theoretical one, which as mentioned shows maxima and
2oc
Percent Titraled
(4) 100%
The compositions of the mixtures are expressed as volume-percent. Wool violet and polyvinylpyrrolidone (PVP) and various commercially available surfactants, including polyethylene glycol 4000. Tween 40 (sorbitan monopalmitate ethyleneoxide adducts) and Tergital (nonylphenol ethyleneoxide adducts) were tested as protective colloids. Apparatus. Scattering measurements were made with a Brice-Phoenix light scattering photometer equipped with a Virtis digital voltmeter. The instrument was calibrated in the usual manner with Ludox ( 1 1 ) . A cylindrical cell, 4-cm diameter, with flat entrance and exit windows, and with the back painted black. was used. A dilute solution of fluorescein was used to check the angular symmetry of the instrument. Titrations, Forty mL of the bromide solution were pipetted into the cell described above. Silver bromide seed, prepared as described below, was added in specified amounts. The silver nitrate solution was added from a 2-mL or 5-mL microburet, with the tip immersed into the solution. The delivery of 0.1 mL titrant. took from 3 t o 5 s. The solution was stirred at 400 rpm during addition and for 10 s after addition. The time interval (s) between addition of increments of titrant is referred t o later as At. M o s t of the scattering measurements were made at 0 = 25' and 40'. M silver nitrate were Preparation of Seed. Tem mL of M popipetted into a 50-mL beaker containing 10 mL of tassium bromide being stirred at 400 rpm. Ten after delivery, 2 mL of the suspension were pipetted into a 100-mL volumetric flask containing 80 mL of isopropanol, and water was added to the mark. The M seed thus prepared was stable, as measured by scattering, for at least 100 h. Electron microscopy using carbon replicas (12) showed that the particles were practically monodisperse and approximately spherical with an average radius of 46 nm, which corresponds t o CY, = 0.9 at A, = 436 nm.
l
OC
Figure 8. A s Figure 7, but 90 % 2-propanol; curve 1, 10-6 M Br-, 10 % seed; curve 2, 5 X lo-' M Br-, 20% seed
minima before EP. The agreement is fair with 10% and good with 100%. Maxima and minima are not observed with 1 % seed because this amount of seed is not sufficient to prevent nucleation during the titration, with the result that the particles do not grow as large as assumed in t h e calculation. This is shown very clearly by comparison of initial and final particle sizes (measured by specific scattering ( 2 ) ) ,given in Table 11. Apparently 100% seed is completely effective in preventing nucleation, and 10% seed much less so. As a practical matter, the use of 10% is preferable to a much larger amount, because the total variation in the measured scattering is larger with the smaller amount. T h e location of the EP is entirely satisfactory in this case. Other alcohols tested include methanol ( 7 0 % ) , ethanol (70-90%), 1-propanol ( 8 0 % ) and tert-butanol (70-9070). Of these the latter two are useful with M Br-, but generally 2-propanol is to be preferred and the further remarks apply to this substance. When specially prepared dust-free water is used instead of ordinary deionized water to prepare the seed and solutions, t h e initial scattering is reduced somewhat but the shape of t h e TC is unchanged. Increase of At to 60 or 120 s causes virtually no change in the shape at 1.04 M Br-. As expected, 8 = 25" gives a better curve (larger slope before EP) than 40". Even 5 X lo-' M Br- can be titrated accurately in 90% 2propanol (Figure 8). The accuracy of the method was tested a t [Br-] = loM5,5 X and 5 X lo-' M a t alcohol contents respectively 30,50,80, and 90%. In 20 such tests, the maximum deviation from the true value was 2.5%, and the precision (standard deviation) was 1.3%, which is scarcely larger than the precision in the buret reading. T h e titration time is less than 5 min. As compared to previous nephelometric procedures (see Ref. 1 and references contained therein), the use of the watermiscible organic solvent and of seed have made a substantial improvement in the accuracy and reduction of the time required for titration of such dilute solutions. The titrations are for obvious reasons much more sensitive than turbidimetric ones, in which transmission is measured, and compare faM level. vorably with any other method a t the LITERATURE CITED (1) E. J. Meehan and Grace Chiu, Anal.
