732
Anal. Chern. 1986, 58,732-734
Solution Phase Complexing of Atrazine by Fulvic Acid: A Theoretical Comparison of Ultrafiltration Methods Donald S. Gamble* Chemistry and Biology Research Institute, Agriculture Canada, Ottawa, Ontario K 1 A OC6, Canada, and Department of Chemistry, Concordia University, 1455 Boulevard de Maisonneuve Ouest, Montreal, Quebec H3G 1M8, Canada Mohammed I. Haniff and Raymond H. Zienius Department of Chemistry, Concordia University, 1455 Boulevard de Maisonneuve Ouest, Montreal, Quebec H3G l M 8 , Canada
Three ultrafiltration methods for solutlon phase complexlng measurements have been reported In the literature. Continuous flow wash out, contlnuous flow wash In, and batch dllutlon curve methods have all been descrlbed operationally In the Ilterature. The batch dilution curve method Is compared to two contlnuous flow methods by mathematical analysls, for the partlcular case of atrazine complexlng equlllbrla in fulvlc acid solutlons. The potential advantages and disadvantages of the three methods have been identlfled for subsequent experimental checklng.
Three types of ultrafiltration methods have been reported in the literature, for the measurement of complexing equilibria in solutions of polyelectrolytes including humic materials. They are the continuous flow wash out method, the continuous flow wash in method, and the batch dilution curve method. Blatt et al. (1)have published integral equations for the first two methods in the absence of complexing. Another two of their integral equations contain correction terms for equilibrium binding of monomeric solute by the ultrafiltration membrane. They have also considered the effect of partial rejection of the low molecular weight solute by the membrane. Grice and Hayes (2) and Grice et al. (3) have described calculation techniques for the continuous flow methods, which employ areas under curves in plots of concentration vs. filtrate volume. The integral equations presented by Smedley (4)for the continuous flow methods have used the Freundlich isotherm for describing surface sorption equilibria. Buffle et al. (5) and Staub et al. (6) have reported the application of a batch dilution curve method to Zn(I1) complexing in fulvic acid solutions. Almost concurrently, Haniff et al. (7) used it for measuring the solution phase complexing of atrazine by fulvic acid. The advantages and disadvantages of this method should be considered carefully, and future experimental work should be guided by a comparison of the three ultrafiltration methods. This will require that the collection of important mathematical equations in the literature be expanded. For each of the three methods, a mathematical analysis is needed, which accounts for solution phase complexing equilibria. The mathematical analyses will be restricted to cases in which ideal membranes fully retain polymeric species and allow free passage of low molecular weight solutes. The purpose of the membrane tests in the other part of this series (8) is to determine experimentally whether or not there are deviations from this ideal behavior. Buffle et al. (5)have also discussed some membrane causes of the deviations. *To whom correspondence should be addressed at Chemistry and Biology Research Institute, Agriculture Canada.
The first objective of this work is to carry out mathematical analyses of the three ultrafiltration methods, within the context of solution phase complexing equilibria. This will permit the characteristics of the batch dilution curve method to be compared with those of the two continuous flow methods. A second objective is to examine the abilities of the three methods to determine both complexing capacities and complexing equilibria.
THEORY General Characteristics of Ultrafiltration Methods for Complexing Measurements. The Complexing Equilibrium. The complexing equilibrium of atrazine by fulvic acid as previously described (7) is given by eq 1.Mo and MOB are the
RB = (l/&) = hfO,/hfOc
(1)
molarities of free and complexed atrazine in the sample. The concentration of fulvic acid, C, is given by eq 2 in which W
c = w/v,
g/L
(2)
is grams of fulvic acid and V, is sample volume inside the ultrafiltration cell. Equation 3 shows total atrazine concenCT
=W
d V C
= Mo
+ MOB
(3)
tration CT, with WT being moles of atrazine. In the theoretical examination of ultrafiltration methods, it is useful to relate total atrazine concentration to that of free atrazine. This is done with eq 4, which has been obtained from eq 1 to 3. The
Cr = (1 + R n C ) M o (4) sample volume within an ultrafiltration cell is calculated with eq 5 , in which Voand V , are initial volume and the volume
vc = vo - v,
(5)
of filtrate. Before ultrafilgration when V, = 0, the initial concentrations of free and complexed atrazine are Moo and MOBo. Equations 2 to 4 then take on the initial values of eq 6 to 8. It should be noted that the amount of fulvic acid C" = (6)
w/vo
CTO
= Wl0/V0 = MoOB MBO
+
(7)
remains constant in these equations, because it has been
0003-2700/86/0358-0732$01.50/00 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
assumed that all of it is retained by the ultrafiltration membrane. General Differential Equation for Atrazine Material Balances. If an atrazine stock solution having a concentration of C, is fed into the ultrafiltration cell from a reservoir, then the amount of atrazine dw, added to the cell is described by eq 9. dV, is the volume of stock solution. With an effluent dw, = C , dV, (9) concentration of C,, the amount of atrazine dwf removed simultaneously into the filtrate is dwf = C, dVf (10) The accumulation of atrazine in the ultrafiltration cell is accounted for by the difference in eq 11. dwT may be ex-
= dw, - dwf (11) panded according to eq 3, as is seen in eq 12. Assuming that dWT = CT dV, + V, dCT (12) uncomplexed atrazine passes freely through the ultrafiltration membrane, then eq 9 to 12 give the relationship in eq 13. ~ W = T CT dV, + V, dCT = C, dV, - C, dVf (13) Equation 13 is the general differential equation for atrazine material balances in ultrafiltration methods. Solutions to the Material Balance Equation for the Three Experimental Method. Continuous Flow Wash Out. Atrazine initially present in the ultrafiltration cell is washed out with a continuous flow of solvent, so that C, = 0. The cell volume is constant at V, = Vo, which gives dV, = 0. Because only free atrazine canpass out into the effluent, C, = Mo. With these values, eq 13 reduces to Vo dCT = -Mo dVf (14) Because the fulvic acid concentration C is constant, eq 4 gives the simple expression for dCT dCT = (1 RBC) d M 0 (15) Equations 14 and 15 together yield eq 16, the differential 1 d In Mo = (16) Vo(1 BC) dVf dWT
+
+
equation for the wash out method. Integrating this over the filtration range 0 to V, liters gives eq 17, for the molarity of
Mo= Mooe-vf/vO(l+RBc)
(17)
uncomplexed atrazine in the filtrate. Mo is also, of course, the molarity of uncomplexed atrazine inside the ultrafiltration cell. Continuous Flow Wash In. The ultrafiltration cell initially contains only a fulvic acid solution, with the constant concentration of C. The atrazine stock solution washed in has the constant concentration C,. The consequencesof a constant sample volume in the cell are that V, = Vo, dV, = 0, and dV, = dVf. Atrazine concentration in the filtrate must again be C, = Mo.These substitutions into eq 13 give Vo dCT = C, dVf- Mo dVf (18) With the use again of eq 15 for the elimination of dCT, rearrangement gives eq 19, the differential equation for wash d In (C, - Mo) 1 -(19) d Vf vO(1 RBC) in. With an initial atrazine concentration of 0, integration of this over the filtration process produces eq 20 for free atrazine concentration.
+
Mo =: c,[1 - gvf/vO(l-RBc)]
(20)
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Batch Dilution Curve. Aliquots in a series are taken from an atrazine stock solution, and each is diluted to the same initial volume Vo in an ultrafiltration cell. Each aliquot therefore produces a batch sample. Atrazine analyses of the filtrates from these batch samples then yield a dilution curve of the original stock solution. For complexing measurements, each batch sample contains dissolved fulvic acid. During filtration nothing is added to the ultrafiltration cell, so that dV, = 0, and -dVf = d(Vo - Vf) = dV,. In the filtrate C, = Mo, as before. In contrast to the continuous flow methods this method does not have a constant cell volume inside the ultrafiltration cell. In addition, it is not known, a priori, how the atrazine concentrations depend on these volumes. The atrazine weight differential dwT is therefore used. These substitutions into eq 13 leave only the two terms seen in eq 21. The relationship of free atrazine concentration Mo to the dWT = Mo
dV,
(21)
weight of total atrazine is obtained by deducing eq 22 from 3 and 4. Equations 21 and 22 now give eq 23, the differential
equation for the method. This uses the fact that the weight d In WT = d In (V, KBW) (23)
+
of fluvic acid, W, is constant. Integration over the volume decrease from Vo to ( Vo - V,) gives the amount of atrazine remaining in the ultrafiltration cell, as calculated from
The amount of atrazine in the filtrate must be the difference between initial and final weights
w = WTo
- WT
(25)
The average concentration of atrazine in the filtrate is defined by Mo = W f / V f (26) The relationship of the experimental quantity Mo to the initial total atrazine concentration within the ultrafiltration cell may now be deduced from equations 24 to 26 and 7. Equation 27 is the result. This should be compared to eq 28, which is the
Mo =
VO
v, + KBWCTO
rearranged form of eq 8.
Moo =
V, .u
v, + KBWCTO
According to eq 27 and 28
Mo = Moo The concentration of atrazine measured in the filtrate therefore equals the initial concentration of free atrazine inside the ultrafiltration cell, before filtration. The experimental result is therefore independent of the volume of filtrate, V,. This assumes, however, that activity coefficient effects are not important. If the concentration and V, are reasonably small, this should be the case.
RESULTS AND DISCUSSION Conclusions from the Solutions to the Material Balance Equation. Table I outlines the derivation of solutions to the general differential equation, for the three methods. The solutions to eq 13 for the two continuous flow methods may be obtained by using volumes and concentrations, because
734
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
Table I. Ultrafiltration Methods for Measuring Atrazine-Fulvic Acid Complexing Equilibria: Solutions to the Material Balance Equation dWT = CT dV, dCT dWT = C, dV, - C, d V
integration limits method
dwT V, dV,
continuous flow wash out continuous flow wash in batch dilution curve
Vo
v, dWT
0 0
c,
du,
c,
dVf upper lower
0 Mo dVf constant dVf Mo dVf constant O Mo -dV,
sample volume and fulvic acid concentration inside the ultrafiltration cell are both constant during ultrafiltration. Because this is not true for the batch dilution curve method however, the differential equation must be set up in terms of the amount of atrazine and the constant amount of fulvic acid inside the ultrafiltration cell. Unlike the solutions to the differential equation for the continuous flow methods, the integral equation obtained for the batch dilution method is not the required final answer. The final result iVoE Moo must instead be deduced from it. The reason for this is that the derivations for the continuous flow methods deal directly with the filtrates, while that for the batch dilution method is concerned with the contents of the ultrafiltration cell. In the absence of fulvic acid or otherwise in the absence of complexing, the terms KBC and RBW vanish. The integral equations then reduce to simpler forms. In the case of the continuous flow methods, these simpler forms are the equations published by Blatt et al. (I), and later presented by Smedley (4). Without complexing, the batch dilution curve method gives the trivial result A?o = CTO, according to eq 27. The equations by Blatt et al., which account for solute binding to ultrafiltration membranes, have forms that look like those of the continuous flow methods with solution phase complexing, in Table I. It is concluded from a comparison of these equations that care must be taken to avoid serious confusion on two points: (a) The equations of Blatt et al. that deal with binding by membranes must be carefully distinguished from those in Table I, which account for solution phase complexing. The behavior of the experimental sample must not be confused with the properties of the measurement system. (b) The equations of Blatt et al. emphasize the need for testing ultrafiltration membranes for interferences. The experimental implications of eq 2 are quite far reaching. In any investigation of labile solution phase complexing, the most general experimental problem is that of finding analytical chemical methods that can distinguish quantitatively between free and total reactant, without disturbing the equilibrium under investigation. If an ultrafiltration membrane shows ideal behavior, with free passage of low molecular weight reactant and total retention of polyelectrolyte substrate both with and without bound reactant, then according to eq 29, the complexing may be measured without the equilibrium being disturbed. Information Obtainable from t h e Three Methods. Grice et al. (3)have noted that an important advantage of the continuous flow experiments is that they may be automated. In addition, it is evident from the continuous flow equations in Table I that the complexing equilibrium function RBcan be calculated directly from the measurements of MOin the filtrate. For the continuous flow wash out procedure, the slope of a logarithmic plot would provide data for ItB,with no value
Mo
Moo Vf
Mo
0
WT
lower
upper
V, Vo- Vf
integral equation Mo = ~
0 0
Vo
o O ~ - ~ f / ~ O ( l ~ ~ B ~ )
M o= C,[l - e-vf/vO(l+c) 2-1
~ ~ " ( [ ( v oVf) - + K ~ w l / ( V o+ KBWI
UT-=
for the constant Moo being required. A single wash out experiment, however, will give only equilibrium data. If capacity were to be determined with this method, it would have to be with a series of experimental runs giving increasing CTo values. In a continuous flow wash in experiment, the concentration MOB of complexed atrazine may be calculated from CT and the integral equation in Table I. CT is known from the experimental design. It can be seen that an experimental curve of MOB vs. Vf should approach the complexing capacity as a limiting value. In this way, a single automated wash in experiment should provide both capacity and equilibrium data. In contrast to the continuous flow methods, the batch dilution curve method cannot be easily automated. This may not be a serious limitation, however. A dilution curve experiment consisting of a series of batch dilutions gives data for both capacity and equilibrium. One of the important implications of eq 29 and 7 is that & may be calculated simply and directly from the measurements of atrazine concentration in the filtrate. Accurate numerical values for CTo, C", and Mo are required, but it is not necessary to measure either Vo or Vf. Each experimental data point, obtained from just one portion of filtrate, produces one Z B value. Logarithmic transformations are not required. These simplifications might serve to reduce the effects of experimental errors in KB Such errors might include chemical analysis errors and interferences from the ultrafiltration membrane. The batch dilution curve method seems to be the most efficient means for testing the membranes for such interferences. In the event that a set of data from the batch dilution curve method shows both random and bias errors, then eq 30 may make the best use of the data. KB =
B1 - BlI1
(30)
The experimental constants B1 and Bal have been described in the preceding paper (8). The evaluation of B , and B,1 by the method of least squares, and the subtraction of bias errors are the reasons why eq 30 might make the best use of the measurements. Registry No. Atrazine, 1912-24-9. LITERATURE CITED (1) Blatt, W. F.; Robinson, S. M.; Bixler, H. J. J . Anal. Biochern. 1968, 26, 151. (2) Grice, R. E.; Hayes, M. B. H. Proc. Br. Weed Control Conf., Tlth 1972, 784. (3) Grice, R. E.; Hayes, M. B. H.; Lundie, P. R.; Cardew, M. H. Chem. Ind. (London) 1973, 233. (4) Smedley, R. J., personal communication, 1984. (5) Buffle, J., Staub, C. Anal. Chem. 1984, 5 6 , 2827. (6) Staub, C.; Buffie, J.; Haerdl, W. Anal. Chem. 1984, 56, 2843. (7) Haniff, M. I.; Zienius, R. H.; Langford, C. H.; Gamble, D. S. Sci. Health, Part B 1985, 820,215. (8) Gamble, D. S.; Haniff, M. I.; Zienius, R. H. Anal. Chem., preceding paper In this issue.
RECEIVED for review July 30,1985. Accepted October 30,1985.
