918
A. SCHWARZ, J. A. MARINSKY, AND E(. S. SPIEGLER
SelfmExchangeMeasurements in a Chelating Ion-Exchange Resin'
by A. Schwarz,2 J. A. Marinsky, and K. S. Spiegler Department of Chemistry, State University of New York at Buffalo, Buffalo l 4 ? New York (Received August 8, 1965)
Self-exchange measurements using NaZ2, CoBo,and Zn66 were made with Dowex A-1 chelating resin, a polystyrene-divinylbenzene copolymer containing the iminodiacetic acid functional group attached to the hydrocarbon matrix. The experimental results indicate that a particle diffusion mechanism is the rate-controlling step in the self-exchange process.
Introduction The preparation of an ion-exchange resin with iminodiacetic acid functional groups to provide chelating properties was reported by Morris, et al.3 Their objective was to produce a resin with sufficient cation specificity to improve chromatographic separations. Such specificity was indeed exhibited by their product.4,6 Its rate of exchange, however, was shown to be somewhat slow, leading to diffuse elution bandsjr6 and the exchange involving one or two strongly chelated cations was reported to be controlled by a second-order chemical reaction.6 This result was unexpected since it was the first observation of a chemically controlled process in an ion-exchange reaction.7 Self-exchange experiments similar to those performed by Boyd and Soldano8 with Dowex 50 were initiated9 to reduce the complexity of the system. In that study it was shown that if the isotopic exchange is controlled by chemical reaction, a three-parameter equation is needed to describe the rate curves. I n this investigation, we have made additional selfexchange studies of N a + and the strongly chelated ions Zn2+ and Co2+ in unbuffered systems and have found that the rate of the self-exchange is diffusion- rather than chemically controlled. Experimental Materials. Dowex Chelating Resin A-1, a product of the Dow Chemical Co., Midland, hlich., was employed in the experimental program. Two different resin preparations were used. The isotopic exchange experiments with sodium were performed a t the Technion-Israel Institute of Technology, with one resin sample source, while the experiments with cobalt and The Journal of Physical Chemistry
zinc were performed a t the State University of New York at Buffalo with the second resin sample source. The nuclides, 5.27-year Co60 and 250-day Zn65, were received from the Radioisotopes Division of the Oak Ridge Sational Laboratory. Carrier-free 2.6-year Na22 was obtained from The Radiochemical Centre, Amersham, England. Reagent grade chemicals were used throughout the program. Resin Preparation. The resin was cycled repeatedly with 2 M HCI and 2 M SaOH. After the sixth conversion to the hydrogen form, the resin was washed with distilled water until the effluent gave a negative test for chloride ion. The resin was then dried at room temperature over Pz06. The dry resin, after separation into four size fractions, 40/60, 60/80, SO/lOO, and 100/120 mesh (U. S.), was stored over Pz05a t the ambient temperature. Preparation for a self-exchange experiment consisted of converting 1.0 g. of the dry resin (2.70 mmoles of iminodiacetic acid) to the sodium form with con(1) Dowex Chelating Resin A-1, a product of the Dow Chemical Co., Midland, Mich. (2) Postdoctoral Research Fellow, 1962-1963: address corresgondence to Oak Ridge National Laboratory, Oak Ridge, Tenn. (3) L. R . Morris, R. A. Mock, C. A. Marshall, and J. H. Howe, J . Am. Chem. SOC.,81, 377 (1959). (4) It. Turse and W. Riemsn, 111, Anal. Chim. Acta, 24, 202 (1961). (5) R . Christell, S. Forberg, and T. Westermnrk. J . Inorg. Nucl. Chem., 19, 187 (1961). (6) R. Turse and W. Riernan, 111, J . Phys. C h m . , 6 5 , 1821 (1961). (7) F. Helfferich, " Ion-Exchange," McGraw-Hill Book Co., Inc., New York, N. Y., 1962, pp. 250-322. (8) G. E . Boyd and B. A. Soldnno, J . Am. Chem. SOC., 75, 6091 (1953). (9) A. Schwarr, Ph.D. Thesis, Technion-Israel Institute of Technology, Haifa, Israel, 1962.
SELF-EXCHANGE MEASUREMENTS IN A CHELATING ION-EXCHANGE RESIN
centrated NaFS08 sodution which was then diluted with a small excess of sodium hydroxide solution. The resin in the sodium form was then washed free of excess NaN03-NaOH solution and an excess of the metal' nitrate solution of the desired concent,ration was added This conversion path was selected since the exchange reaction is more rapid and proceeds further towardl completion when the resin is originally in the sodium1 rather than the hydrogen form. Equilibrium of the resin with the metal nitrate solution was assured by replacing the external solution three or four times, after a contact time for at least 24 hr. After the first solution change, no observable loss of metal from the external solution was detectable and the pH of the solution usually remained constant. The chelated metal in the resin was then tagged with1 its radioactive isotope by equilibration for a t least 1 week with a metal nitrate solution containing the radioactive nuclide. This method ensured attainment, of equilibrium between resin and solution. Particle Size Measurement. The diameter of the wet resin beads of a given mesh size in equilibrium with the desired solution was measured by microscopic examination with a calibrated eye piece ocular which had been previously standardized against a sitage micrometer. Measurement of 300 to 400 beads was employed t o obtain a representative average particle size. Experimental Arrangement. A schematic representation of the apparakus for monitoring the isotopic exchange is shown in Fig. 1. A four-necked 250-ml. flask was used to contain the resin and solution in a thermostated bath. A stirrer extended into the mixture through the center neck of the flask. A filter stick of coarse sintered glass confined the resin in the flask during continuous circulation of the solution (by means of a peristaltic pump) through a Plexiglass flow cell situated in the crystal well of st shielded scintillation counter attached to a recording count-rate
919
meter. The flow all was held firmly in the well by tape, thus assuring constant geometry. Experimental Procedure. In a typical experiment, tracer-free metal ion solution of the same concentration as that used for the resin equilibration was pumped in and through the system until thermal equilibrium was attained. The pumping rate was sufficiently high to assure circulation of the entire solution at least once every minute. Self-exchange was initiated by introducing the wet radioactive resin through the fourth flask neck (Fig. 1) into the circulating solution, followed by rinsing with a small quantity of the metal salt solution. Uniform mixing of the solution and resin particles was achieved by vigorous stirring. The volume of the final solution was adjusted to make the ratio between total exchanging ion in the resin and that in solution (w) equal to 0.1. With this ratio of the radioactive isotope will eventually enter the solution. Progress of the self-exchange process, as indicated by the increasing activity in the solution, was followed by the recording rate meter. A self-exchange reaction was considered to be complete when no change in solution activity was noticeable in a 24-hr. period. A radioactive material balance was periodically carried out to check on the attainment of equilibrium. The background was measured a t the completion of each run by removing the flow cell from the scintillation counter and by recording the rate meter reading while the resin and solution were still in place. The time constant of the rate meter was increased in the final solution activity measurement to give 1% standard deviation. Because of the importance of this final activity value for the accurate estimate of the fractional exchange, a separate check on instrument stability was made by recording the activity of a Co60 reference source before and after representative runs. (No correction for instability was ever needed, however.)
--
Results
4 . Gornrno Scintillotion detector 2 . Plexi-glass flow cell 3. Circulating pump 4. Coarse sintered glass 5 Stirrer controlled by thyratron
0 /POWERSUPPLYI*]
-1
RATE METER
THERMOSTAT
Figure 1. Schematic rep:resentation of experimental arrangement for study of isotopic eschange.
The results of the self-exchange experiments with NaZ2,GoBo,and Zn@are summarized in Tables I-IV. If the self-exchange is controlled by ion diffusion in the resin, the equations shown in Table IV apply. The conventional test for this condition is a plot of Bt, as determined from the experimental values of fractional attainment of equilibrium by this formula, against time. If this yields a straight line through the origin, it is proven that B remained indeed constant during the experiment, as demanded by the theory. An additional criterion is the variation of the slope of the line inversely with the square root of the radius, To, Volume 68,Number Q
April, 1964
A. SCHWARZ, J. A. MARINSKY, AND K. S. SPIEGLER
920
Table I : Self-Exchange of Sodium in Dowex A-1" at 29.0 f 0.1" p H in
Solution
soln.
0.08 M NaAc 0.1 M NaOH 0.1 M
Particle radius, om. X 102
3.2 7.6 12.6d
Halfexchange,
Rate meter time const., sec
SBC.
1.17 1.43 1.44
34 5 9
3 1 1
.
W
0.01 0.1 0.2
Water contentb
D x 107, cm.2 0ec.-1
0.52 1.37 1.72
1.2 11 6.7
0 Self-exchange studies with XaZ2 were made with a different batch from those reported for Zn and Co. * Grams of water per gram of dry resin in the H form. c Solution also contained 0.05 M acetic acid. pH uncorrected for sodium error.
~~~~~~~~
Table I1 : Self-Exchange of Zinc and Cobaltous Ions in Dowex A-1" in 0.1 M Metal Nitrate Solutions a t 29'
Expt. no.
Wet radius, cm. X 10%
Apparent diffusion Halfcoefficientb exchange, Dapp X log, min. cm.2 set.-'
Znss 1 2 3 4 5 6
0.874 f 0.05 1.05 f 0 . 0 9 1.32 f 0 . 0 9 1.76 f 0 . 1 4 1.32 1.32
pH 5 , 3 f 0 . 3 6.2 8.4 17.0 28.4 9.5 11.9
1.05 1.29 1.69
6.4 6.3 5.1 5.4 9.1 7.4
0 . 1 M Zn(NO8)z 0 . 1 M Zn(NO& 0 . 1 M Zn( NO& 0 . 1 M Zn(NO& 0 . 1 M ZnSOn 0.2 M Zn(NO3)z w = 0.05
pH 4 . 6 f 0 , I
COB0 7 8 9
Remarks
9.0 11.6 19.1
6.1 7.4 7.5
Resin water content = (38.5 f 1.4)7, water by weight in the zinc form and 42.5% water by weight in the cobalt form. * See Appendix for explanation and definit'ion of this term.
~
Table 111: Reproducibility of Self-Exchange of Zn in Dowex A-1 in 0.1 M Metal Nitrate Solution at 29"
Expt. no.a
Wet radius, cm. X 102
3 10 11 6b 6f 6gb 1
1.32 1.32 1.32 1.32
IC
0.874 0.874
Halfexchange, min.
17.0 14.5 18.5 19.2 33.0 8.9 6.2 9.0
Apparent diffusion coefficient,
D~~~ x 109, om.2 sec.-l
5.1 5.9 5.0 4.6
6.4 4.2
a An experiment number without a letter (column 1) corresponds to a fresh sample. Experiments designated by a number followed by a letter were carried out by recharging a used sample. Consecutive letters refer to consecutive experiments. * In this experiment the sample was treated with acid and base prior to use.
Q
of the resin spheres The results in Fig. 2 demonstrate that both criteria were met in the self-exchange of zinc, at least throughout the major part of the reaction. A similar test applied to the self-exchange of sodium9 and cobalt also leads to the conclusion that the self-exchange of these ions is particle-diffusion controlled. In these calculations, the values of Bt as functions of F were read from Reichenberg's table,1° which applies when the ratio, w, of total metal ion in the resin to that in solution is close to zero; in this case the rate of the reverse reaction (re-absorption of radioactive ion into the resin) may be neglected. This table was used in all experiments in which w was 0.1 or smaller. For the evaluation of all those experiments in which w was larger than 0.1, however, we used a table due to Boyd (ref. 7, p. %5), based on a more complicated rate The Journal of Physical Chemistry
formula, which takes due account of the reverse exchange reaction. The diffusion coefficient values that are presented in Table I were calculated from one point between 50 and 60% exchange. Since the self-exchange of Naf is relatively fast, the rate meter was operated at a low time constant setting and the statistical deviation was 5% at the half exchange; the over-all accuracy for the diffusion coefficient is believed to be &lo%. The results indicate that this exchange is particle-diff usion controlled. Rieman and Turse also found the same mechanism in exchange reactions in which one of the ions was sodium. The increase of the diffusion coefficient with increasing water content of the resin in the first two experiments in Table I can be correlated with enhanced rate of diffusion in the less tortuous path (10) D. Reichenberg, J. Am. Chem. Soc., 7 5 , 589 (1953). (11) G. E. Boyd, A. W. Adamson, a n d L. S. Myers, Jr., ibid., 69, 2836 (1947).
921
SELF-EXCHANGE MEASUREMENTS IN A CHELATING ION-EXCHANGE RESIN
EX
Figure 2. Infinite bath diffusion plot for self-exchange of zinc between Dowex A-1 resin of different particle radius, T O , and 0.1 cm., expt. no. 1; M Zn(NOa)zsolutions: 0 , ro = 0.874 X A, ro = 1.05 X 10-2cm., expt. no. 2; 0 , T O = 1.32 X cm., expt. no. 4. cm., expt. no. 3; 0, T~ = 1.76 X -
Table IV : Numerical Evaluation of the Self-Diffusion Coefficient for a Representative Experiment" t, see.
43.2 115.2 211.5 369 560 927 1440
F
Bt
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.0386 0.0928 0.177 0.301 0.479 0.734 1.120
B, RC.-~
x
104
8.94 8.06 8.37 8.16 8.55 7.92 7.78 8 . 2 5 =t0 . 7 (av.)
Solution, 0.103 M Zn(?i03)*; resin, Dowex A-1 in the zinc form; average particle radius, TO = 8.74 X 10-8 cm.; w = 0.1. Infinite bat>hequation used to calculate the apparent diffusion coefficient from the fractional attainment of equilibrium, F, at time 1
B
E
D,,,+/roa
Numerical values of Bt as a function of F are from Reichenberg.Io
available in the more swollen resin. These two sodium exchange reactions involve the first ionized carboxylate group of the imino diacetic acid and the comparison is straightforward. In the third experiment, however, the geometry of the system is probably changed by the presence of the second ionized carboxylate group.
The resin becomes a less open structure as shown by the lower water content per sodium. atom at higher pH. Although the total water content of the resin is higher, the structure is more restrictive to movement of the sodium ion. The diffusion coefficient values reported in Table I1 for Zn2+ and Cot+ are each an average of seven points. An example of a detailed numerical evaluation of the self-diff usion coefficient is given in Table IV. From experiments 1, 2, 3, and 4 with zinc, and experiments 7,8, and 9 with cobalt, the strong dependence of the exchange rate on particle size is clearly evident. The half-exchange time changes from 6 min. for the smallest particle examined to 28 min. for the largest particle in the case of Zn and from 9 miii. to 19 miii. for Co. On the other hand, a single diffusion coefficient cm.2 see.-' with average values of (5.7 f 0.7) X sec.-l for zinc and cobalt, and (7.0 f 0.9) X respectively, describes the complete exchange reaction. By comparing experiments 3 , 10, and 11 (Table 111)) we can judge the reproducibility of the experiments. These experiments were performed with new samples from a prepared resin source. Apparent diffusion cocm.2 sec.-l were efficient valuesof (5.7 f 0.7) X always obtained. However, when an experiment was repeated with the same sample by recharging the resin with fresh radioactive tracer, the rate of the self-exchange always decreased. This phenomenon is illustrated by the results obtained from experiments 1 and IC and from experiments 6 to (if. When trying to repeat experiment 1, after carrying out an experiment a t a lower temperature, the rate v7as observed to decrease (experiment le). In experiments 6b to 6f, a single sample of resin was exchanged in five runs including a run at 50°, and the decrease of exchange rate was quite large. It is believed that poor chemical and physical stability of Dowex A-1 is responsible for this difficulty. The Dowex A-1 batch used to carry out the selfexchange study was only 1% DVB cross linked.12 Also, it has been shown13that a polymeric mixture is leached from the low cross-linked resin into the equilibrating solution. This leach material may block diffusion through the pores and thus decrease the exchange rate. In an attempt to support this hypothesis, the resin sample after experiment Gf was regenerated and cycled twice between 2 dl HC1 and 2 M SaOH solutions. After (12) R. M. R h e n t o n , T h e Dow Chemical Co., private communicntion, 1961. (13) C. Eger a n d J. A. Marinsky, private communication, 1963.
Volume 68,niumber 4
April, 1964
922
this treatment an additional run was made. The halfexchange time dropped to 8.9 min. A microscopic examination of the beads a t this stage revealed that a high percentage was broken, as compared to 95% sphericity of the fresh resin sample. The water content of the sample after this experiment was also higher (43%) than the average value reported in Table 11. It is of interest to note that a sodium hydroxide treatment is also recommended14 by the Dow Company when the resin loses its exchange ability due to long storage in the hydrogen form. A possible explanation for this recommended procedure is the opening of the structure and removal of trapped soluble material when the resin is converted to the sodium form. No further attempt was made to investigate this breakdown of the resin. Despite this difficulty, freshly prepared samples gave repeatable diffusion coefficients. The resin decomposition is apparently much slower than the self-exchange process so that by taking a fresh sample for each experiment, meaningful diffusion coefficient values are obtained. By changing the salt solution, the effect of concentration (Table 11, expt. 6) and nature of the co-ion (Table 11, expt. 5 ) was studied. When the external solution concentration is doubled, the rate increases by about 20%. It is believed that the increase in counter-ion diffusion coefficient with the increase in external solution concentration is due to electrolyte invasion as observed with other resins.15 I n Dowex A-1 this effect will be magnified when the resin is loaded with a chelate-forming metal ion because of the decrease of Donnan exclusion due to the metal-ligand bond. A pronounced co-ion effect is found; self-exchange is much faster with an external solution of 0.1 I d ZnSO4 than with 0.1 M Zn(NO&. The sulfate ions are known to form ion pairs with zinc and about 55% of the Zn is estimated to be bound to l6 at the concentration of the experiment. Thus much of the diffusible zinc in the resin pores is in the form of the noncharged ZnS04 molecule, and one species instead of two or three species as in the case of Zn(NOs)zprevails in the resin phase. Both effects increase the rate of exchange.
Discussion The theory of ion-exchange kinetics is well-established. Evidence in the extensive literature7 indicates that the rate-determining step is interdiffusion of the exchanging counter-ions either in the particle (particlediffusion mechanism) or in a film of solution surrounding the resin sphere (film-diffusion mechanism). Furthermore, diffusion exchange was also found to be a The Journal of Physical Chemistry
A. SCHWARZ, J. A. MARINSKY, AND K. S. SPIEGLER
rate-controlling step in a different chelating resin studied by H0jo.l’ The rate equations for homogeneous reaction were inapplicable to the absorption of Ag, Hg, and Cu by (3-hydroxyphenyl)thiourea-phenol resin. The diffusion of ions in the resin phase seemed to be the rate-determining step in this case. We conclude from our results that the self-exchange of Zn and Co in Dowex A-1 is also diffusion-controlled in contradiction to the results of earlier work6 with this resin. We believe that the lack of dependence of exchange rate on particle size observed by Turse and RiemanB in their earlier work was due to a composite of complicating effects resulting largely from the difference between exchange of different ions and of isotopic ions. The analogy between heat transfer and ion exchange which leads to the basic equations for particle-diffusion controlled exchange is justified oiily for exchange of ions of equal mobility and charge, as in isotopic exchange experiments, and not for other cases (ref. 7, p. 269). Moreover, swelling and deswelling during the exchange reactions of Turse and Rieman may have further complicated their exchange kinetics and obscured the effect of particle size variation on the rate. Additional experimental evidence on which Turse and Rieman base their earlier conclusion is the dependence on the external solution concentration. This observation, however, does not exclude a diffusion process,15especially in a system where electrolyte invasion can become significant. Finally, recent work by Rieman and VaronI8 also supports particle-diffusion control as the rate-limiting factor. Most complexes of Zn2+and Co2+are known to be labile, i.e., exchange rapidly with isotopic free ions, and there is no apparent reason for them to become inert when chelated in a resin when only three bonds need to be broken simultaneously. The iminodiacetic acid group in Dowex A-1 is connected to the polymer chain through a methylene bridge and is therefore relatively free to oscillate. It might be expected that a truly chemically controlled self-exchange reaction will be encountered in the case where the metal ion is inert ( e . g . , Ni2+ and Cr3+). Work on Cr513 + self-exchange has been initiated and preliminary results indicate that the exchange may be chemically controlled. (14) “Dowex Chelating Resin A-I,” The Dow Chemical Co., Midland, Mich., 1959. (15) R . Schlogl, Z . Elektrochem., 57, 195 (1953). (16) R . A . Robinson and R. H. Stokes, “Electrolyte Solutions,” Academic Press, Inc., New York, N. Y., 1955, p. 402. (17) N. Hojo, J . Chem. SOC.Japan, Ind. Chem. Sect., 62, 1145 (1959); Chem. Abstr., 57, 10756 (1962). (18) W-.Rieman, 111, private communication, 1963.
923
SELF-EXCHANGE MEASUREMENTS IN A CHELATING ION-EXCHANGE RESIN
Acknowledgment. Financial support through Contract XO.AT(30-1)-2269 with the U. s. Atomic Energy Commission is gratefully acknowledged.
Appendix Self-Diffhsion i n a Traced Chelating Resin. Consider the internal dissociation M+
M+R-
+ R-.
+
(2) is The asterisk refers to the radioactive species. the total (stoichiometric) concentration of the radioactive ion, bound and free, in the resin (mole C*&{+ t h econcentration of the free radioactive ions only. Because of the equilibrium (1), the chemical potential of the radioactive compound 11+R-is equal to the sum of the chemical potentiah of the respective ions
+
P*M+
PR-
(3)
and since M R - is independent of position within the resin particle, because there is no concentration gradient of R- within the particle &*M+ ___ - dP*MR dx dx ~
(4)
By definition p*MR
= ( p * ~ ~ .f ) o RT
In Z*MR =
+ RT 'In fC*MR
(P*MR)o
(5)
Note that the concentration, C*MR,in the resin is almost identical with the total stoichiometric concentration of nI*, in the resin. Remembering that the activity coefficient, j,is not a function of x d p * ~ ~RT
_ _ _ --_ _ _dC*.t __
dx
C*at dx
or, if we define
we obtain
(1)
where &I+ is thie metal ion and R - the resin anion. The effect of electrolyte invasion is neglected. This equilibrium is assumed to be instantaneous. Theorem. The self-diffusion flux of' the metal ions can be treated a,s if there were no chelating or other binding between RI+ and R-; however, the apparent self-diffusion coefficient, D,,,, calculated from this procedure is smaller than the true self-diffusion coefficient of M+, D M . I n fact, D a p p= ~ D MCY ,beingthe ratio of free M + l,o total R l (M+RM+) in the resin. Prooj. The basic flux equation for jree ions AT+ in the resin is
P*MR=
(7)
(6)
According to eq. 4 this expression equals dp*M +/dx. Substituting it in eq. 2, we obtain
(9) The meaning of eq. 9 is that for the self-diffusion flux of free ions, M+, in the resin, one does not have to use the Fick law for free ions (whose concentration, after all, one does not know, because the degree of dissociation and the extent of invasion are unknown). Instead one can use, in Fick's law, the total stoichiometric concentration of radioactive ions. The apparent diffusion coefficient obtained when this is done is simply the true ionic diff usion coefficient multiplied by the fraction of the free ions in the resin. The same treatment applies when electrolyte invasion is not entirely negligible, provided the ratio of invading ,If + to the total (stoichiometric) amount of M in the resin is small, which was no doubt the case in our experiments, because dilute electrolyte solutions were used.
Symbols All symbols refer to the resin phase. 6 * M R activity of radioactive R'I+R-, mole cm.-3 B constant defined in Table IV, sec.-l C*M + concentration of free radioactive 11+ ions, mole C*MR concentration of radioactive M+R€*st concentration of radioactive M+R- plus free radioactive XI + Dapp self-diff usion coefficient obtained from Fick's law +RI is when concentration gradient of radioactive ;\ used in that law, sec.-l DM true ionic self-diffusion coefficient of A I + obtained from Fick's law when gradient of radioactive A I + is used in that law f activity factor of M+RF fractional attainment of equilibrium TO resin radius, cm. R gas constant, joule mole-' deg.-' t time, sec. T absolute temperature, OK. w the ratio of total metal in the resin to that in solution a t the end of a self-exchange experiment. Its numerical value is determined by the equilibrium distribution of the tracer between resin and solution x length coordinate, cm. CY ratio of free ions h l + to total AI in the resin
Volume 68,Number Q
A p r i l , l.904
924
MORTO~L. WALLACH AND WILFRIED HELLER
flux of radioactive RI+, mole cm.-2 see.-' chemical potential of radioactive ions 11+, joule mole-1
+*>I-
chemical potential and standard chemical potential of radioactive M +R-, respectively ~ - chemical potential of R-
~ * M R ,(p*nir~)o
p * -~
p
Experimental Investigations on the Light Scattering of Colloidal Spheres. VI.
Determination of Size Distribution Curves by
Means of Turbidity Spectral
by Morton L. Wallach and Wilfried Heller Department of Chemistry, Wayne State Unicersitzl, Detroit, Michigan
(Receizled August IS, 1968)
Spectra of turbidity exhibited by polystyrene latices were investigated and the results were used to derive the size distributions in these systems. The turbidity spectra method was found to be equivalent in performance to that based upon the study of spectra of the scattering ratio (depolarization) provided the spectral turbidity maximum is within or near the spectral range investigated. Turbidity spectra were determined for systems showing a negatively skewed, a positively skewed, and a gaussian-type distribution. The distribution curves derived from the spectra were compared with electron microscopic distribution curves obtained on the same systems. Simplifications of the method proposed here are outlined which may be sufficient if the objective is merely a determination of the modal and/or mean particle diameter or if the concentration of the scattering material is not exactly known.
f(r)
Introduction The preceding paper in this series2 dealt with a method for determining size distribution curves in heterodisperse systems of colloidal spheres by means of the spectra of the scattering ratio. An alternate method may be based upon the use of turbidity spectra. The theory of this alternate method was given some time ago.3 The present paper is concerned with the experimental test of this alternate method, using again heterodisperse polystyrene latices as model systems. The distribution curve is again assumed to be of the type The Journal of Physical Chemistry
= (y - ~ , , ) ~ - I ( ~ - ~ ~ ) / ~ l a =
0, r
(1)
< ro
Here, Cf ( T ) dr is the number of particles per unit volume of a system containing particles with radii between r and r d?, 5 is a parameter proportional to the width
+
(1) This work was supported by t h e Office of Naval Research. T h e results given in the present paper were presented at the 134th Sational Meeting of the American Clheniical Society, Chicago, Ill., Sept., 1958. (2) W. Heller and M.L. Wallach, J . Phus. Chem., 67, 2577 (1963). (3) M. L. Wallach, W. Heller, and A. F. Stevenson, J . Chem. Phys., 34, 1796 (1961).
