Environ. Sci. Technol. 2002, 36, 3815-3821
Analysis of Trace Metal Humic Acid Interactions Using Counterion Condensation Theory RODOLFO D. PORASSO,† JULIO C. BENEGAS,† M A R C A . G . T . V A N D E N H O O P , * ,‡ A N D SERGIO PAOLETTI§ Department of Physics, IMASL, National University of San Luis, 5700 San Luis, Argentina, National Institute of Public Health and the Environment, P.O. Box 1, 3720 BA Bilthoven, The Netherlands, and Department of Biochemistry, Biophysics and Macromolecular Chemistry, University of Trieste, 34134 Trieste, Italy
Recent extensions of counterion condensation theory, originally developed for well-defined linear polyelectrolytes, enable us to analyze the interaction of trace metals with humic acid. In the present model, the heterogeneity of the macromolecule is taken into account as well as the chemical binding of the considered metal ions to the humic material. Experimentally, potentiometric titrations have been performed for humic acid in solution in the presence of different environmentally important (heavy) metals (Ca, Cd, Cu, Ni, and Pb) at various metal concentrations by titrating with potassium hydroxide without additional salt. From proton release data obtained for the initial point in the titration, it was estimated that the interaction of the different metals with the humic acid in terms of binding strength increased in the order Ca < Cd ≈ Ni < Pb ≈ Cu. These results were confirmed by model analysis. Experimentally obtained apparent dissociation constants were in good agreement for the humic acid systems containing Ca, Cd, and Ni at concentrations ranging from 0 up to 0.75 × 10-3 mol L-1 and polymer dissociation degree from about 0.1 up to approximately 0.8. Also for the Cu/humic acid and Pb/humic acid systems, the agreement between experimental data and calculated data was satisfactory at the lowest metal concentrations over the complete titration curve. For elevated levels of Cu and Pb, the agreement between experimental data and theoretical calculations becomes less satisfactory at low degrees of dissociation of the humic acid. This distortion of the potentiometric curves is probably due to changes in the intrinsic pK of the functional groups due to metal binding. This complex process is not included in present polyelectrolytic models.
Introduction Knowledge of the interaction of heavy metals with organic complexing agents in natural aquatic systems is of great importance in order to understand the mobility and the (bio)availability of the metals in these systems. This subject has * Corresponding author phone: +31-30-274-4013; fax: +31-30274-4441; e-mail:
[email protected]. † National University of San Luis. ‡ National Institute of Public Health and the Environment. § University of Trieste. 10.1021/es010201i CCC: $22.00 Published on Web 08/01/2002
2002 American Chemical Society
therefore become an important study area for basic and applied scientists (1-8). Heavy metals are persistent elements some of which, like zinc and copper, are essential for organism. On the contrary, “at elevated levels” their presence may lead to toxic effects. Nowadays, it is well recognized that the speciation of trace metals in natural aquatic solutions, i.e., the distribution over different physicochemical forms, plays a key role in the understanding of the ecological effects both in terms of deficiency as well as toxicity. The speciation of trace metals is highly affected by the presence of complexing agents such as humic and fulvic acids, which in general can be described as (large) heterogeneous macromolecules, charged due to the dissociation of different ionisable groups under natural solution conditions. Although the interaction of metal ions with rather simple complexing agents such as small organic and inorganic anions (e.g., EDTA and chloride) is quite well understood, the behavior of trace metals in the presence of large natural macromolecules is still difficult to predict. One of the problems is related to the chemical structure of these complexing agents. Since these natural macromolecules are formed by the random condensation of degradation products of plants and animals, their composition strongly depends on their origin (9); hence, uniform models to describe their precise chemical structure are not available. A number of authors have reported that these polymers are constituted by several types of functional groups, like carboxylic and phenolic ones (10-13). The ionization characteristics of these macromolecules play an important role in understanding their interactions with heavy metal in natural environments. For this reason, the acid-base properties of natural humic material have been studied quite extensively, resulting in proposed descriptions of chemical properties such as abundances of constituting functional groups (10-13) and affinity interactions with different metals (1, 7). In general, it can be said that, because of the polyacid features of the macromolecule, its charge increases with increasing values of pH resulting, for the case of metal interactions, in an increased fraction of associated trace metals. Exact chargepH functionalities appear to be dependent on the intrinsic pK values, the distribution of ionizable groups, and the presence of various counterions. Finally, natural aquatic systems usually consist of a mixture of metal ions of different nature; hence, competitive effects should be taken into account in describing the distribution of metal ions over free and associated states. A number of models have been presented in the literature for describing metal/humic acid interactions. Some examples of those models are the NICA-Donnan (5), the fully coupled electrostatic (3), Model V (11), polyelectrolyte (14, 15), and oligoelectrolyte (16). In Kinniburgh et al. (5), a comparison of these models, including how they deal with heterogeneity and electrostatics, is reported. Recently, we have extended the counterion condensation theory of linear polyelectrolytes, first presented by Manning (17), to describe potentiometric titration data of a potassium/ humic acid system in water under salt-free conditions as well as in the presence of supporting simple 1:1 salt (18). In this approach, it was shown that the model consistently describes the interactions of the potassium counterions with the heterogeneous macromolecule over a wide range of ionization degrees (charge densities) of the humic acids and covering a salt concentration range of 2 orders of magnitude. Previous work has furthermore shown that, besides the polyelectrolytically associated (condensed) counterions, trace metals might also be bound chemically to specific sites on VOL. 36, NO. 17, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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the macromolecule (19). This process results in a modification of the polyelectrolyte overall charge density and, consequently, in the polyelectrolytic effects that are controlling the actual speciation in solution. Polyelectrolyte systems containing various metal ions (Cd, Cu, and Pb) in solution with highly charged synthetic polyelectrolytes such as poly(acrylic acid) and poly(methacrylic acid) were successfully described by the model (19). With this background, we aim with this work to study how the specific interactions of different heavy metals with a natural heterogeneous polyelectrolyte humic acid modify the polyelectrolytic characteristics of the mixed metal ion solutions. Experimentally, potentiometric titrations of humic acid in the presence of the metal ions Ca2+, Cd2+, Cu2+, Ni2+, and Pb2+ at different metal concentrations were performed. It will be shown that use of the above model in the analysis of the experimentally obtained data leads to the determination of polymer characteristics and metal speciation.
Experimental Section Materials. The humic acid was a commercial sample from Fluka (Lot/Product No. 19582067/53680). A detailed description of the pretreatment procedure of the humic material is reported in our previous work (18). In short, to obtain the soluble fraction of the humic acid, 10 g of the Fluka sample was dissolved in 1 L of water. First, the pH of the solution was increased up to a value of 9 and then reduced to a value of 3. The obtained suspension was centrifuged to separate the solid and dissolved humic material. The supernatant, containing the dissolved fraction, was then dialyzed against Millipore water (refreshing daily the dialyzate). The dialyzed humic acid solution was subsequently treated with an ion exchanger to transfer the material into the acid form. The final solution had a dissolved organic carbon content of 80 × 10-3 mol (L of C)-1. The pH and the conductance of this solution were 2.87 and 590 µS cm-1, respectively. The concentration of chargeable groups in the final stock solution as determined by conductometric titration (20) was found to be 16.2 ( 0.3 × 10-3 mol L-1. The stock solution was stored in the dark at approximately 7 °C until usage. Titrisol potassium hydroxide solutions (Merck), at a concentration of 0.100 mol L-1, were used as titrating solutions. Cadmium nitrate, calcium nitrate, copper nitrate, and lead nitrate were obtained from Merck. Nickel nitrate was obtained from Analar. Tritisol buffer solutions (Merck) were used to calibrate the potentiometer. All chemicals were of analytical-reagent grade or better. Water was obtained from a Millipore reverse-osmosis system and degassed with nitrogen before use. Methods. The potentiometric measurements were performed using a digital pH meter from Radiometer (model PHM 95) in combination with a glass/calomel electrode (GK2401C). The electrode system was daily calibrated using two buffers (pH 4 and 7) and controlled with a standard solution of pH equal to 6.88. The potentiometric titrations were performed with potassium hydroxide charging the metal/humic acid system at an equivalent fixed concentration of chargeable groups of 2.50 × 10-3 mol L-1 and at different concentrations of metal ions (i.e., 0.25 × 10-3, 0.50 × 10-3, and 0.75 × 10-3 mol L-1). The pH readings ((0.01 pH unit) were carried out after 6 min of stabilization. All the experiments were performed in duplicate in a glass vessel thermostated at 25 °C ((0.3 °C) under a nitrogen atmosphere to avoid introduction of CO2 in the solution. Duplicate measurements were all found to be within 0.05 pH unit from each other. The degree of humic acid dissociation (R) was experimentally determined from the monitored pH value and the degree of neutralization using 3816
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FIGURE 1. Acid-base potentiometric titrations of humic acid at four concentrations of Cd(NO3)2: 0 (0), 0.25 × 10-3 ()), 0.50 × 10-3 (4), and 0.75 × 10-3 (O) mol L-1. Humic acid concentration ) 2.50 × 10-3 mol L-1.
R)
[KOH]a + [H+] - [OH-] Cp
(1)
where [KOH]a is the added concentration of potassium hydroxide solution, corrected for dilution effects, and Cp is the concentration of humic acid solution in terms of functional groups (thus including both dissociated and not dissociated groups).
Results and Discussions Potentiometric Titration Curves. In Figure 1, the experimental pH values are plotted versus the degree of dissociation of humic acid at total equivalent polymer concentration of chargeable groups of 2.50 × 10-3 mol L-1 in the presence of different Cd(NO3)2 concentrations (ranging from 0.25 × 10-3 to 0.75 × 10-3 mol L-1). For reference purposes, a titration without added metal has been included. The following features can be observed: (i) For all curves, there is a large increase in pH between the starting point of the titration and the highest value of dissociation reported in this work. This observed increase in pH largely overcomes the usual change in pH shown in the ionization of linear weak polyelectrolytes, even those of high charge density (21) like poly(acrylic acid) or poly(methacrylic acid), which are often used as model polyelectrolyte systems for humic acids. (ii) The large change in pH overshadows the usual ionic strength effect, i.e., the smaller change in pH for titrations performed at higher ionic strengths. (iii) Various regions of different convexities may be identified, a feature that has been previously attributed to titrations of polyelectrolytes constituted by different types of functional groups, as is the case of humic acid (11, 22, 23). (iv) The initial titration point (none KOH added), corresponding to the self-dissociation of the polyelectrolyte, is different in the four sets of data points. It is shifted to higher values of R and lower values of pH for higher metal concentrations. In Table 1, these initial values of pH and R are summarized for all studied metal/humic acid systems. For the reference titration of humic acid (i.e., in the absence of any trace metal), an initial pH of 3.67 and a corresponding initial R value of 0.086 are obtained. It is clearly seen that these initial values of pH and R appear to be dependent on the type and concentration of the metal ion present in the titrating solution. For the same metal concentration, the initial pH increases in the order Cu ≈ Pb < Ni ≈ Cd < Ca, while the initial value of the degree of ionization R increases in the
TABLE 1. pH and r Values of the Initial Metal/Humic Acid Solution (No KOH Added)a Ca2+ metal 2+ [Me ] pH r 0 0.25 0.50 0.75
3.67 3.55 3.52 3.49
0.086 0.113 0.121 0.128
Cd2+
Ni2+
Cu2+
Pb2+
pH
r
pH
r
pH
r
pH
r
3.67 3.52 3.46 3.42
0.086 0.121 0.137 0.150
3.67 3.52 3.48 3.46
0.086 0.119 0.131 0.139
3.67 3.44 3.36 3.30
0.086 0.144 0.173 0.198
3.67 3.47 3.36 3.25
0.086 0.136 0.173 0.222
a In all cases [HA] ) 2.50 × 10-3 mol L-1, and the total metal concentration [Me2+] is reported in units of 10-3 mol L-1.
FIGURE 3. Experimental (symbols) and theoretical (lines) pKa vs r values for the Cd/humic acid system. Model calculations are performed with ∆Gb ) -12 RT. For symbols and lines see Figure 2.
FIGURE 2. Experimental (symbols) and theoretical (lines) pKa vs r values for the Ca/humic acid system. Ca(NO3)2 concentrations: 0 (0), 0.25 × 10-3 ()), 0.50 × 10-3 (4), and 0.75 × 10-3 (O) mol L-1. Humic acid concentration is 2.50 × 10-3 mol L-1 and ξ ) 0.8. Model calculations are performed with ∆Gb ) -9 RT; lines refer to metal concentrations: 0 (s), 0.25 × 10-3 (- -), 0.50 × 10-3 (- -), and 0.75 × 10-3 (- - -) mol L-1. reverse order. For these experimental conditions, the release of protons has been calculated from the difference in initial pH of the reference system and the systems with added metal. For all metals, it is found that the amount of released protons is much more pronounced than in the case of potassium nitrate solutions (19). This observation cannot be explained on polyelectrolyte effects solely and certainly indicates that the association of the metal ions with the humic acid is different for the different studied metals. On the basis of these results, the metal binding strength to humic acid is expected to be the weakest for Ca and the strongest for Pb for the present metal/humic acid systems. Apparent Dissociation Constants, pKa. Further analysis of the titration data is carried out by comparison of the experimentally determined and theoretically calculated change of the apparent dissociation constant (pKa) with the degree of dissociation (R). In Figures 2-6, the experimentally obtained apparent dissociation constant is plotted as a function of R for the Ca/humic acid, Cd/humic acid, Ni/ humic acid, Cu/humic acid, and Pb/humic acid systems, respectively. In all cases, the equivalent concentration of humic acid chargeable groups is 2.50 × 10-3 mol L-1, and the metal concentrations are 0 (no added metal), 0.25 × 10-3, 0.50 × 10-3, and 0.75 × 10-3 mol L-1 without other additional salt added. The experimentally pKa values have been calculated from the measured pH using:
FIGURE 4. Experimental (symbols) and theoretical (lines) pKa vs r values for the Ni/humic acid system. Model calculations are performed with ∆Gb ) -13 RT. For symbols and lines see Figure 2.
(2)
FIGURE 5. Experimental (symbols) and theoretical (lines) pKa vs r values for the Cu/humic acid system. Model calculations are performed with ∆Gb ) -15 RT. For symbols and lines see Figure 2.
The more relevant feature in all cases is that the titration curves show a very large increase in pKa (∆pKa ≈ 5), far larger
than what is calculated in any theoretical approach due to polyelectrolytic effects in the ionization of monoprotic polyelectrolytes. This large increase together with the pres-
pKa ) pH + log
[1 -R R]
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FIGURE 6. Experimental (symbols) and theoretical (lines) pKa vs r values for the Pb/humic acid system. Model calculations are performed with ∆Gb ) -15 RT. For symbols and lines see Figure 2. ence of regions of different convexities have been shown to be characteristic of potentiometric titrations of multifunctional (heterogeneous) polymers, as is the case of humic acid. Moreover, for the Ca/humic acid system, the curves for the various metal concentrations have approximately the same shape but start at lower values of pKa (and consequently higher values of R) when the metal concentration is increased. The main differences between the titration data for the different metal ions appear at the beginning of the titration, i.e., for R e 0.25. The shifting in initial pKa and R is greatest for the Cu/humic acid and Pb/humic acid systems. A similar observation was found in previous work, in which we analyzed, both experimentally and theoretically, acid-base properties of the well-defined linear polyelectrolytes poly(acrylic acid) and poly(methacrylic acid) in the presence of Ca, Mg, Zn and Cu (19). Furthermore, for the Cu/humic acid and Pb/humic acid systems, the experimental data at high metal concentrations are strongly distorted at the beginning of the titration, highly depending on the concentration of the metal added. This feature was also observed for Cu/ polyacrylic and Cu/polymethacrylic systems (19). Modeling. The present model is developed within the framework of counterion condensation theory and takes into account (i) polyelectrolytic interactions, (ii) chemical heterogeneity of the macromolecule, (iii) chemical binding of counterions, (iv) ionic strength effects, (v) entropic effects, and (vi) competitive polyelectrolytic interactions between counterions of different valences (18, 24, 25). The main points of the theory are presented in the Appendix. In the following analysis, the experimentally obtained potentiometric titration data are compared with the change in pKa calculated from eqs A5 and A6. These calculations are presented as continuous lines together with the experimental data in Figures 2-6 for the Ca/humic acid, Cd/humic acid, Ni/humic acid, Cu/humic acid, and Pb/humic acid systems, respectively. As pointed out above, the large increase in pKa together with the presence of regions of different convexity is characteristic of potentiometric titrations of multifunctional (heterogeneous) polymers. A detailed analysis of the reference curve of the same humic acid (i.e., in the absence of any added metal) has been performed in detail by Porasso et al. (18). The structural polyelectrolyte charge density of the humic acid determined in that work is ξstr ) 0.8 and will be used throughout the present calculations. This value seems to be quite realistic considering estimations reported in the literature for the distance between two consecutive sites on different humic material assuming a local cylindrical geometry (26). Other structural parameters characterizing the 3818
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humic acid like the number and fractional abundances (Xi) of the functional groups together with their intrinsic pK0 values (pK0i ) in salt-free solutions have also been taken from Porasso et al. (18) and are included in Table 2. From Figures 2-6, it can be seen that, in general, the theoretically determined values for pKa are in good agreement with the experimentally obtained ones, especially for all Ca/ humic acid, Cd/humic acid, and Ni/humic acid solution systems. Note that model calculations have been carried out without modifying the previously found (18) structural parameters of the humic acid, i.e., keeping the fractional abundances of the four types of functional groups constant for all the different metal ions and metal concentrations of the present study. In the case of Cu/humic acid and Pb/humic acid solutions, the agreement between theoretical calculations and experimental data is reasonable for low metal concentrations. At higher metal concentrations, the model does not fully reproduce the experimental results because of the strong distortion of the titration curves. This feature is probably due to the strength of the binding of Cu and Pb to humic acid, which results in important changes in basic polyelectrolyte characteristics such as the intrinsic pK0 values of the remaining ionizable groups after (strong) binding of counterions in their surrounding. If that is the case, further theoretical work is needed in order to describe properly this part of the experimental data. To obtain the calculated curves at all different metal concentrations, only two parameters are varied: (i) the pK0i values of the four functional groups and (ii) the value of the binding free energy, ∆Gb. The quality of the simulation (e.g., the agreement between the experimentally obtained and theoretically calculated values) is determined by the square root of sum of the squares of the differences between calculated and experimental pKa values divided by is the number of experimental points in a given titration, i.e.,
x∑(pKtheor - pKexp)2/N. The corresponding values are in-
cluded in Table 2. From Table 2, it can be seen that the pK0i values of the first two groups show a small but systematic decrease when the concentration of divalent ions is increased. The magnitude of this shifting in the pK01 and pK02 values reaffirms the ordering of the Me/humic acid specific interaction: Ca < Ni ≈ Cd < Cu ≈ Pb. This binding strength feature is correspondingly reflected in the theoretical calculations through the value of the intrinsic binding free energy, ∆Gb, which runs from -9 RT for the Ca/humic acid system to -15 RT for the Pb/humic acid and Cu/humic acid systems. On the other hand, the pK0i values used for the third and fourth groups remains essentially constant in all calculated curves. This seems to indicate that (at the metal concentrations of the present work) the binding of metal ions occurs mainly to the functional groups with the two lower pK0i values. One should nevertheless consider that the third functional group has the lowest fractional abundance, and consequently, its influence in the calculated curves is smaller. Metal Speciation. Of particular interest for the chemical and environmental characterization of the interaction process of heavy metals with humic acid is the determination of the distribution of heavy metals over the different polymer/metal ion association states. In our previous work (24), we have defined two modes of polymer-metal association: polyelectrolytically driven counterion condensation and chemical bonding. The first mode is a loose territorial confinement of the counterions to a volume in the immediate vicinity of the polyelectrolyte (the condensation volume). The second mode is a site-binding process that responds to a mass-law equilibrium. From our modeling, the contribution of these two modes to the associated metal population can be
In all cases the humic acid concentration ) 2.50 × 10-3 mol L-1, and the mean charge density ξ ) 0.8. N refers to the number of experimental data points in a given titration curve. a
2.4 4.3 6.4 9.0 21 0.065 2.4 5.1 6.4 9.0 21 0.061 2.4 4.4 6.4 9.0 21 0.053 2.7 4.7 6.4 9.0 21 0.040 2.9 5.3 6.4 9.0 22 0.027 3.1 5.0 6.4 9.0 21 0.046 3.1 5.2 6.4 9.0 20 0.053 3.3 5.3 6.4 9.0 21 0.020 3.0 4.9 6.4 9.0 21 0.037 3.2 5.0 6.4 9.0 21 0.045 3.3 5.4 6.4 9.0 20 0.036 3.3 5.0 6.4 9.0 21 0.040 3.4 5.3 6.4 9.0 19 0.045 3.6 5.3 6.4 9.0 18 0.045 3.7 5.4 6.4 9.0 23 0.017 0.19 0.29 0.17 0.35 N Σ(pKtheor - pKexp)2/N
X1 X2 X3 X4
mol
3.0 5.3 6.4 9.0 22 0.035
0.75 0.50 0.25 0.75 0.50 0
0.25
0.50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
0.25
-15 -13 -12 -9 0
(10-3
L-1) ∆Gb (RT)
[Me2+]
pK01 pK02 pK03 pK04
-15
Pb2+ Cu2+ Ni2+ Cd2+ Ca2+ none metal
TABLE 2. Values of Intrinsic pK0i and Fractional Abundances Xi of Functional Groups of Humic Acid As Determined from Model Analysis of Potentiometric Titrationsa
TABLE 3. Effect of Metal Binding on Effective Charge Density, ξσ ) ξ(1-2 σ), of Humic Acids at Different Total Metal Concentrations for Ca and Cu metal concentration
0.25 × 10-3 mol L-1
0.50 × 10-3 mol L-1
0.75 × 10-3 mol L-1
Ca/humic acid Cu/humic acid
0.489 0.480
0.338 0.320
0.235 0.160
calculated. The first task is to determine if the polyelectrolyte effective charge density is higher than the critical value, ξcrit. If the effective charge density remains below this value, only chemical bonding occurs; otherwise, counterion condensation also takes place (24). Taking into account the average amount of (calculated) bound Me2+ per ionizing group (σ) one can determine the mean charge density after metal binding, ξσ ) ξstr(1 - 2σ). As an example, the calculated values of the effective charge density, ξσ, for the highest ionization degree reached in these experiments (R ) 0.80, corresponding to an uncompensated mean structural charge density ξ ) Rξstr ) 0.64) are presented in the Table 3 for the two extreme cases of humic acid/metal binding interaction, i.e., for Ca and Cu. In these examples, the striking change in polyelectrolyte effective mean charge density due to the chemical binding of the heavy metal ions is readily observed. Model calculations following the mixcounterion condensation model (25) indicate that in all cases the effective charge density remains below the critical (threshold) value, i.e., no counterion condensation of either monovalent or divalent counterions is found under the present experimental conditions. Therefore one can safely conclude that only chemically bound heavy metal ions are associated to the humic acid. Their concentrations are readily calculated at each point of the titration curve by using the expression Cb2 ) σRCp. Ionization Dynamics of the Functional Groups. As a further step in the study of polyelectrolyte-metal interactions, the model can be used to calculate the association of the metal ions and the ionization of the different types of functional groups as a function of the degree of polymer ionization. The theoretical values of the concentration of bond divalent counterions (Cb2 ) σRCp) and the concentration of ionized monomers (the binding sites) of the different functional groups (Cion ) βiCi) can be straightforwardly i calculated as a function of R. The calculations have been done assuming free binding energies of ∆Gb ) -9 RT (representing the case of Ca/humic acid) and ∆Gb ) -15 RT (representing the case of Cu/humic acid), for the three metal concentrations experimentally studied. These are plotted in Figures 7 and 8, respectively. The concentrations of ionized functional groups (the charged sites) have been calculated for the case of the lowest metal concentration. For higher metal concentrations, there are small insignificant differences because of the small changes in pK0i . It is seen that the concentration of counterions chemically bounded depends strongly on the free energy of binding and on the concentration of the divalent counterion. For the case of ∆Gb ) -9 RT (representing the Ca/humic acid system), at the three total metal concentrations and over all the R values presented here, the concentration of bound counterions is lower than the concentration of the first and second ionized functional groups. In particular for R < 0.65, the only type of functional groups ionized are of the first ones, usually attributed to carboxylic type. In Figure 8, similar curves obtained for the titration in the presence of stronger binding metal ions (∆Gb ) -15 RT; representing the case of Cu/humic acid) are presented. The effect of the stronger interaction between polymer and metal ions is clearly observed. Calculations show that in this case all metal ions available in solution should VOL. 36, NO. 17, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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to be of “phenolic” type) starts above R ≈ 0.6. Consequently, chemical association of the metal ions of the present study to monomers of this type of group should be very small under the presently employed experimental conditions.
Acknowledgments R.D.P. is a doctoral fellow of CONICET and is grateful to RIVM for financing his stay at the Institute, where the experimental part of this work was carried out. M.A.G.T.v.d.H. is grateful to the Universidad Nacional de San Luis (Argentina) for funding his stay as a visiting professor to the university. This work has been partially supported by CONICET (Argentina).
Appendix FIGURE 7. Model calculations of the concentration of bound metal ions (symbols) and of the ionized concentration of four functional groups (lines) for increasing values of dissociation and different total metal concentrations using ∆Gb ) -9 RT and ξ ) 0.80 representing the experimentally studied Ca/humic acid system. Data for Xi and pKi0 are summarized in Table 2. Lines represent: 1st group (s), 2nd group (- -), 3rd group (‚‚‚), and 4th group (-‚-). Total metal concentrations: ()) ) 0.25 × 10-3, (O) ) 0.50 × 10-3, and (0) ) 0.75 × 10-3 mol L-1.
The usual starting point of counterion condensation (CC) theory (27) is to model the polyelectrolyte as a uniform linear array of charges. The fundamental assumption is that the total polyelectrolyte charge can be smoothly spread over the polymer and that it is possible therefore to define a (mean) structural charge density, ξ, given by
ξ)
lB e2 ) b kTb
(A1)
where lB is the Bjerrum length, b is the average distance between consecutive charges projected onto the polymer axis, e is the elementary charge, is the bulk dielectric constant, k is the Boltzmann’s constant, and T is the temperature. Chemical binding of counterions of valence zj to the polyelectrolyte is considered by reducing the average charge density effectively by an amount equivalent to a fraction σ per polymeric charge (24):
ξσ ) ξ(1 - zjσ)
(A2)
where σ ) [Me]b/Cp, and [Me]b is the concentration of bound metal ions. The effective charge density thus becomes the central characteristic of the polyelectrolytic solution. Following the procedure described in refs 24 and 25, one can write analytical expressions for the polyelectrolytic (Gpol), entropic (Gentr), and binding (Gb) contributions to the total (excess) free energy of the system: FIGURE 8. Model calculations of the concentration of bound metal ions (symbols) and of the ionized concentration of four functional groups (lines) for increasing values of dissociation and different total metal concentrations using ∆Gb ) -15 RT and ξ ) 0.80 representing the experimentally studied Cu/humic acid system. Symbols and lines as in Figure 7. be chemically bound at R ) 0.6, at least for the total metal concentrations reported here (C2 ) 0.25 × 10-3, 0.50 × 10-3, and 0.75 × 10-3 mol L-1), and certainly for very low metal concentrations usually observed in natural environmental systems. The calculated curves also indicate that binding occurs almost completely before that the fourth functional group starts to ionize. As mentioned before, the abundance of the third group is rather small as compared to the ones of the first and second group. This supports the assumption that, under the present experimental conditions at least, the first two functional groups are mainly responsible for the polymer-metal ion chemical association and, therefore, partially justify, a posteriori, the use of only one binding free energy to model the binding process. It is seen that even in the simulated case of the Cu/humic acid system for the highest metal concentration (i.e., 0.75 × 10-3 mol L-1) over 99% of metal ions are bond under these conditions. The ionization of the fourth functional group (often considered 3820
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Gtot ) Gpol + Gentr + Gb ) Gion + Gb
(A3)
Appropriate derivatives of eq A3 yield the functional form of the thermodynamic functions of interest (25). The apparent dissociation constant, pKa, is in general given by
pKa ) pK0 + ∆pKa
(A4)
where pK0 is the intrinsic pKa characteristic of the (isolated) ionizing group making up the polymer and the change in pKa due to the ionization of the polyelectrolyte is included in the term ∆pKa. Once the ionic excess free energy function (Gion) is known, we can readily calculate the following:
∆pKa )
∂Gion 1 ) G(R,ξ,Cp,C1,C2,T,,σ0) (A5) np2.303RT ∂R
where np is the number of polymeric charge units; R is the gas constant; C1 and C2 stand for the analytical concentrations of monovalent and divalent counterions, respectively; and σ0 is the maximum (mean) fraction of chemically bounded counterions per polymeric charge unit, i.e., the mean stoichiometry of the binding process. For a heterogeneous weak polyelectrolyte constituted by N functional groups of fractional abundance (Xi) and intrinsic
pK (pK0i ), the overall intrinsic pK0 is a function of the degree of ionization. Porasso et al. (19) have shown that
pK0(R) ) pK0i + log
[
βi
(1 - βi)
]
1-R R
(A6)
where βi stands for the ionization degree of the ith functional group.
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Received for review July 31, 2001. Revised manuscript received March 26, 2002. Accepted May 23, 2002. ES010201I
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