Environ. Sci. Technol. 2009, 43, 7465–7471
Modeling the Effect of Surface Heterogeneity in Equilibrium of Heavy Metal Ion Biosorption by Using the Ion Exchange Model WOJCIECH PLAZINSKI* AND WLADYSLAW RUDZINSKI Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Cracow, Poland, and Department of Theoretical Chemistry, Faculty of Chemistry, UMCS, pl. M. Curie-Sklodowskiej 3, 20-031 Lublin, Poland
Received March 30, 2009. Revised manuscript received July 27, 2009. Accepted August 4, 2009.
This paper is focused on the theoretical modeling of the heavy metal ions biosorption process. The origin of the commonly observed, so-called “heterogeneity effects” has been explained. Two popular models were considered for that purpose: the adsorption model and the ion exchange model. TheCondensationApproximationmethodhasbeenusedtodevelop the model describing the influence of surface heterogeneity on biosorption equilibria. As this model has been derived by accepting the ion exchange processes on the biosorbent surface, the obtained expressions are able to also take into account the pH effect. The mentioned “heterogeneity effects” may be treated as the result of (i) the existence of the continuous function expressing the binding site heterogeneity; (ii) the existence of a few different kinds of binding sites, having different values of ion exchange constant; (iii) stoichiometry of the ion exchange reaction occurring on the biosorbent surface. The latter case is responsible for the so-called “apparent” heterogeneity, when the adsorption model and the ion exchange model offer equally good fits of data but the degree of the surface heterogeneity estimated in this way might be different.
Introduction Biosorption is an ability of certain types of biomass (living or nonliving) to bind various types of pollutants by mechanisms which are not metabolically driven (1). The biosorption techniques are applied effectively in effluent treatment processes mainly for removal of heavy metal ions (1–4) and dyes (5, 6); other toxic chemicals can also be treated in this way (7). Among advantages of the biosorption processes are cost-effectiveness, efficiency, minimization of chemical/ biological sludge, and regeneration of biosorbent with possibility of metal recovery. Researchers investigating numerous biomass types proposed many useful biosorbents including algae (2, 3, 8), fungi (4, 9), bacteria (10, 11), and others. A variety of mathematical models have been proposed to represent biosorption equilibrium in batch systems and to quantitatively describe the binding of heavy metals to different biosorbents. Most often the biosorption equilibrium data are modeled by mathematical equations adopted from * Corresponding author tel: +48-81-537-5519; fax: +48-81-5375685; e-mail:
[email protected]. 10.1021/es900949e CCC: $40.75
Published on Web 09/01/2009
2009 American Chemical Society
the adsorption models. The following sorption isotherm equations are commonly used to correlate experimental data: Langmuir, Freundlich, Langmuir-Freundlich, Toth, Temkin, Redlich-Peterson, and others (11–13). Nevertheless, as noted by Volesky, these models are just “mathematical functions”, and they hardly reflect the actual biosorption mechanism (13, 14). None of these equations incorporates pH effects, for instance, and the description of the influence of other factors, such as ionic strength, requires applying much more complicated models (15, 16). The issue related to the conclusions which can be drawn from application of the Langmuir adsorption model when the actual process is an ion exchange has been considered by Crist et al. (17). These authors have compared the classical adsorption model on an energetically and chemically uniform surface with the process based on the ionic exchange in which the sorption is accompanied by the release of replaced ions. As mentioned, among the equations used for correlation of biosorption data, many are based on the model of adsorption by a heterogeneous surface (i.e., LangmuirFreundlich equation, Toth equation, etc.). Let us call the set of these models “heterogeneity-related models” (18). A discrete distribution function can also be assumed when one deals with a surface having a few different kinds of adsorption (binding) sites. It seems to be necessary to perform theoretical studies similar to those by Crist et al. but focused on the model of a heterogeneous surface. The following reasons for that can be pointed out: 1. Numerous papers on biosorption of heavy metal ions do not contain a sufficient data body to state whether an adsorption or an ion exchange model should be applied. 2. Equations developed by assuming adsorption on a heterogeneous solid surface and derived originally for the gas/solid systems, have commonly been used for theoretical description (or at least correlation) of data measured in the biosorption systems. 3. A large number of papers have reported on a better applicability of these equations than on the Langmuir isotherm equation. This effect is especially interesting as some of these equations are only two-parameter ones, similar to the Langmuir equation (19–22). Thus, the advantage of these equations lies not only in a larger number of best-fit parameters introduced. The aim of the present study was to find what is the source of good applicability of the heterogeneity-related models to biosorption data and what conclusions can be drawn from this fact. The explanation of this issue may result in better understanding of the process and, in turn, serve for improvement of sorption performance prediction by using mathematical models.
Theoretical Section As a starting point we consider a model based on the ion exchange mechanism which has been recognized as the predominant one for many important biosorption systems (1–3, 15, 17). The following factors will be discussed as potentially responsible for inducing the behavior approximated well by the heterogeneity-related models: 1. The stoichiometry of ion-exchange reaction occurring during metal binding is different from 1:1, predicted by the Langmuir model (compare (23) for more details). 2. The binding site heterogeneity modeled by a continuous distribution function of the ion-exchange constant values. VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Ion Exchange Isotherm Equations for the Three Stoichiometric Assumptions (a) reaction
(b) model equation
κ
A+ + BX T AX + B+ or
NsκCA CB + κCA
(2b)
(CB)2 + 4NsκCA - CB√(CB)2 + 8NsκCA 8κCA
(3b)
qA )
(2a) κ
A2+ + BX T AX+ + B+
κ
A2+ + 2BX T AX2 + 2B+
(3a)
qA )
qA ) κ
A3+ + 3BX T AX3 + 3B+
(4a) -
1 N + 3 s
21/3CB3 9[9Nsκ2CA2CB3 + √κ3CA3CB6(4CB3 + 81N2s κCA)]1/3
[9Nsκ2CA2CB3 + √κ3CA3CB6(4CB3 + 81N2s κCA)]1/3 9·21/3κCA
(4b)
3. The existence of different kinds of binding groups on the biosorbent surface, modeled by a multidiscrete distribution function of the ion-exchange constant values (24). The generalized ion exchange reaction between the dissolved ion A (of valence +R) and the bound ion B (of valence +β) can be written as follows: κ
βAR+ + RBXβ T βAXR + RBβ+
(1)
where X denotes the monovalent binding site, and κ is the ion-exchange equilibrium constant. More details related to the mathematical principles of this model can be found in the Supporting Information (SI). Table 1 contains the mathematical expressions corresponding to reaction 1 obtained for the most commonly assumed stoichiometries and expressed as qA(CA, CB) functions (qA and qB are the adsorbed amounts of ions A or B, respectively, expressed in [mol/g] whereas CA and CB are the equilibrium concentrations of A and B in the liquid phase (in [mol/L])). Three reaction equations, 2a, 3a and 4a, collected there correspond to biosorption of mono-, bi-, and trivalent metal ions by protonated biosorbent (i.e., these are reactions of protons replaced by metal cations; then B+ ) H+ and CB ) 10-pH). For reaction 2a, it is possible to distinguish 1:1 binding stoichiometry for both mono- and bivalent metal ions (the mathematical expression 2b describing the equilibrium remains unchanged). Such an assumption is in agreement with some recent experimental results (25). Reaction 2a and corresponding eq 2b are essentially identical with the mathematical form of the Langmuir isotherm equation; the difference lies in the theoretical interpretation of the Langmuir constant (17). It is also worth noting that all equations presented in Table 1, as well as general eq S10 (in the SI), are only two-parameter equations. Equations 2b-4b were derived with the assumption that each ion A binds to R monovalent sites to form the AXR complex. It differs from the model proposed by Schiewer and Volesky who assumed formation of A1/RX complexes (24). Our present studies will be focused on the model expressed 7466
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by eqs 1-4a, but we postpone a theoretical comparative analysis of these two competitive approaches to our future publication. Let us postulate the existence of a differential, normalized to unity, distribution of the number of binding sites among their corresponding ion exchange constant values κ, which we denote as χ(ln κ) and call the “ion exchange constant distribution” (IECD) function. Then, the averaged metal ion uptake is expressed as follows: qA(CA, CB) )
∫ q (C , C , ln κ)χ(ln κ)d ln κ Ω
A
A
B
(5)
where Ω is the physical domain of ln κ. Further, qA(CA,CB,ln κ) is the so-called “local” isotherm describing the amount of ions bound by the sites characterized by the constant κ. It can be calculated from eq S10 by accepting the stoichiometry appropriate for a given physical system. Three examples of such isotherms are collected in Table 1. The mathematical form of qjA(CA,CB) function depends on both the binding stoichiometry and the accepted IECD function. To arrive at simple, analytical expressions, the Condensation Approximation (CA) method can be applied as a tool for simplifying the r-h-s of eq 5. The principles of this approach as well as the way of deriving eq 6 are presented in the SI. The mathematical formalism developed there can be briefly described as follows: 1. Replacing the exact qA(CA,CB,ln κ) function by the stepwise function called “condensation isotherm”. 2. Substitution of the result into the integral at the r-h-s of eq 5. 3. Eliminating the additional parameter (i.e., the lower limit of integration) by expressing it as the function of CA and CB. After applying the above-mentioned method to eq 5 one obtains: qA(CA, CB) )
Ns R
∫
∞
ln κ0
χ(ln κ)d ln κ
where ln κ0 is expressed as a function of R and β:
(6)
[ (
ln κ0 ) ln CA-βCBR
Ns√β R
3/2
)(
+ R√β
·
β
3/2
)] -R
Ns√R
β
+ β√R
(7)
Let us also note that the attempts to apply the CA method to describe the ion exchange processes on inorganic sorbents were made by Inglezakis (26). In the literature treatment on gas-solid adsorption, it is well-known that, despite a variety of adsorption systems which we may have to deal with, their features can, to a first approximation, be represented by only a few adsorption energy distributions (18). A similar situation is expected to occur also in biosorption systems as the model expressed by eqs 6 and 7 arose from the related assumptions. The following two IECD functions were considered by us: the Gaussianlike function defined by eq 8 and the rectangular function given by eq 9:
(
)
ln κ - ln κc exp n 1 χ(ln κ) ) n ln κ - ln κc 1 + exp n χ(ln κ) )
{
[
(
)]
2
(8)
1 , for ln κmax e ln κ e ln κmin ln κmax - ln κmin 0, elsewhere (9)
where κc, n, κmin, and κmax are parameters (κc is the mean value of κ, while κmin and κmax are its lowest and highest values, respectively; the n value is proportional to the variance of distribution (8)). These two functions have been recognized as those connected with the most commonly applied, heterogeneity-related expression (i.e., with the LangmuirFreundlich and Temkin equations) (18). The essential mathematical features of eqs 8 and 9 are described elsewhere (18). The second way of introducing heterogeneity effects into the model of ion exchange is by the concept of multidiscrete distribution function of the κ value. Similarly to eqs 8 and 9, this function also describes the binding site heterogeneity. This approach is essentially equivalent to accepting the model assuming the existence of different kinds of binding groups on the biosorbent surface. Such models and examples of their applications are known in the literature (25, 27). The simplest case of such a model (which was considered by us) is described in the SI. To obtain the desired qA(CA, CB) (or qB(CA, CB)) solution, the values of the fractions of binding sites of a given type and values of κ characteristic of these sites must be known.
Results and Discussion The results presented here can be divided into three groups: (i) the direct connection between the “heterogeneity-related” adsorption model and the newly proposed equations originating from the ion-exchange model; (ii) the investigation on the effects of the existence of a few different kinds of binding sites (having different values of ion-exchange constant); (iii) the investigation on the “apparent” heterogeneity effects being the result of the ion-exchange driven process of biosorption with the reaction stoichiometry different from the simplest, i.e., 1:1. Finally, a detailed analysis is presented on the basis of the experimental data found in the literature. The applicability of the heterogeneity-related models (expressed by such formulas as the Langmuir-Freundlich, Freundlich, or Temkin equations) for the biosorption equilibrium description can be explained not only in terms of adsorption models. It appears that the mathematical form of the equilibrium isotherm equations (developed in the
previous section) is analogous to the equations developed previously for the model of adsorption onto heterogeneous solids. The newly developed ionic exchange model, however, also takes into account both the surface reaction stoichiometry and the concentration of the replaced ion. Let us focus on the two above functions, which can explain the applicability of the Langmuir-Freundlich (LF) (called also the Sips equation) and Temkin equations to describe biosorption equilibrium. The mathematical forms of these two equations can be represented by the following eq 10 (LF) and eq 11 (Temkin): qA(CA) )
K′LFCA1/h 1 + KLFCA1/h
)
Ns KLFCA1/h R 1 + K C1/h LF A
(10)
qA(CA) ) KT′ + KTln CA
(11)
in which KLF, K′T, and KT are coefficients whose interpretation is dependent on the assumed model. Here we limit ourselves to a small number of simple ionic exchange reactions whose equations are collected in Table 1 as eqs 2a, 3a, and 4a. Table 2 contains the isotherm equations calculated from eq 6 corresponding to these stoichiometries and to eq 8 and eq 9. The functions obtained for general stoichiometry (expressed by R and β parameters) are mathematically identical to the classical LF and Temkin equations (see eqs 10 and 11). The theoretical interpretations of the KLF, KT, and K′T constants resulting from this comparison are shown in Table 3. One has to deal here with a similar situation to the case of the interpretation of the Langmuir isotherm in terms of the ion exchange concept, i.e., the coefficients which should be constant according to the adsorption model (here: KLF, KT, and K′T) are dependent on the concentrations of the exchanged ion B in the solution (17). There is also one interesting issue related to the theoretical interpretation of the 1/h coefficient. From the theories of adsorption equilibria it is known that in the case of LF eq 10 the value of 1/n can vary from 0 to 1 for physical reasons (18). In our model of ion exchange reaction 1 the theoretical interpretation of the exponent of CA is β/n. Thus, in the case of β > 1 and 1/n close to unity one may arrive at the previously mentioned denominator value higher than 1 which cannot be explained on the grounds of the adsorption model. Such values were found by applying the best-fitting procedures in the case of some biosorption systems (28, 29). Further, the heterogeneity effects might be related to the existence of a few different kinds of binding sites which have also different values of ion-exchange constant. The principles of such an approach and models originating from this assumption are well-known in the literature (see also the SI). Here, it is proven that the classical Langmuir-Freundlich model simulates very well the behavior predicted by such a multidiscrete-distribution model, despite the fact that the mathematical forms of equations corresponding to them are different. The original numerical procedure was proposed to compare these two models; its details can be found in the SI. The LF isotherm equation was chosen as a fitting formula due to its mathematical features which reflect: (i) the physical limit when CA f ∞; (ii) the degree of surface heterogeneity can be easily estimated by the value the 1/h coefficient; (iii) it is able to describe adsorption on a homogeneous surface (when 1/h f 1). Thus, the issue we have to deal with can be reformulated into the following question: to what extent can the heterogeneity-related adsorption models approximate the behavior predicted by the ion-exchange model combined with the multidiscrete distribution of κ value (represented by eq S12 in the SI)? VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 2. “Averaged” Ion Exchange Isotherm Equations Calculated from Eqs 9 and 10 reaction
(a) model equation for IECD 11
qA )
5a
qA ) 6a
κ1/n c +
qA ) 7a
κ1/n c +
(b) model equation for IECD 12b
Nsκ1/n c (CB /CA)1/n + κ1/n c
( (
Nsκ1/n c /2 1 + √2 4
Nsκ1/n c /3 1 + √3 243
( )
)
1/n
)
CB2 NsCA
( )
2/n
CB3
qA )
CA CB
(12b)
ln κmax - ln κmin
[ ( ) ( )]
(13b)
) ( )]
(14b)
4κmax CANs + ln CB2 1 + √2 2(ln κmax - ln κmin)
qA )
1/n
( )]
Ns ln
(13a)
1/n
[
Ns ln κmax + ln
(12a)
(14a)
[(
Ns ln qA )
N2s CA
27κmax
+ ln
CAN2s
CB3 6 + 6√3 3(ln κmax - ln κmin)
TABLE 3. Theoretical Interpretation of the Coefficients Appearing in the Temkin 11 and LF 10 Equations coefficient
KLF
interpretation
[ (
κ1/n CB-α/n c
)( β
α3/2 + α√β
·
)]
-α -1/n
Ns√α β3/2 + β√α
(15)
(16)
β/n
1/h
KT
Ns√β
[ (
)(
Ns Ns√β Ns√α · 3/2 ln κmaxCB-α 3/2 R(ln κmax - ln κmin) α + α√β β + β√α
K′T
A few selected results of our studies are shown in Figure 1 and described in the SI. Here only the most important observations are mentioned. (i) The nearly perfect agreement between the ion exchange model (assuming the existence of two different values of κ) and the LF equation is always observed and these two models cannot be separated from
Nsβ R(ln κmax - ln κmin)
β
)] -α
(17)
(18)
each other while having only qA vs CA data at our disposal. (ii) Heterogeneity-related effects are dominant in the case of high differences between the κ1 and κ2 values. The “heterogeneity parameter” 1/h value may drop to about 0.75, which corresponds to a moderate degree of surface heterogeneity. (iii) When the differences between the assumed κ values are
FIGURE 1. Results of fitting the (CA/CB; qA) data generated by using the eq set (S12) by the LF eq 10. The values of accepted parameters are K1 ) 0.1, K2 ) 0.9 (two values of ion-exchange constant), Ns ) 0.002 mol/g. Fractions of both type of sites were equal to 0.5. The units were omitted for the sake of clarity. 7468
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FIGURE 2. Results of fitting the (CA/CBr; qA) data generated by using eq 6 by the LF eq 10. (CA/CBr)max is the maximum value of the CA/ CBr ratio for single sorption isotherm. The values of accepted parameters are K ) 0.5, Ns ) 0.002 mol/g. The units were omitted for the sake of clarity.
FIGURE 3. Fits of the experimental equilibrium isotherms of (a) Cu2+, (b) Cd2+, and (c) Pb2+ ions biosorption by the calcium alginate, reported by Papageorgiou et al. (30). The solid lines (______) are the theoretical isotherms calculated by using eq 19. The values of the obtained best-fit parameters are collected in Table 4. Other details can be found in the text.
FIGURE 4. Fits of the experimental equilibrium isotherms of Cu2 by seed husks, reported by Boucher et al. (31). The solid lines (______) are the theoretical isotherms calculated by using eq. set 19. The values of the obtained best-fit parameters are collected in Table 4. Other details can be found in the text. not high, the LF equation may serve as a formula estimating the concentration of the binding sites (see Figures 1a and S3a). The heterogeneity effects observed when applying adsorption models (represented by the Langmuir-Freundlich equation, for instance) may be also the results of the ionexchange driven process of biosorption with the reaction stoichiometry different from the simplest, i.e. 1:1. In this case such “apparent” heterogeneity can be observed even in the case of homogeneous surfaces. The simplest case when R ) β ) 1 is mathematically identical to the Langmuir model (i.e., to the model of adsorption on a homogeneous surface), thus, it can not simulate the mentioned behavior (17).
The other most popular stoichiometries, i.e., those for which β ) 1 and R ) 2 or 3, were considered. The model equations corresponding to them are collected in Table 1 as eqs 3b and 4b. The procedure details of the comparative procedure are described in the SI. The most important observations are the following (more details can be found in the SI): (i) The nearly perfect agreement between the ionic exchange model expressed by eqs 3b and 4b and the adsorption model expressed by the LF equation is always maintained. This means that the observed differences are not significant enough to distinguish between these two models on the basis of the analysis of the independent sets of experimental data recorded as qA(CA, CB). (ii) The concentration of binding sites, estimated from the best-fit procedure, can vary from Ns/R to Ns, where Ns is the “real” value, which was accepted during generation of the data points. The “calculated” Ns values become close to the above-defined “real” ones when the concentration of ion A in the solution is much higher than that of ion B. (iii) The “heterogeneity parameter”, i.e., 1/h, depends strongly on the CA/CBR ratio value. The behavior characteristic of ideal homogeneous surface (1/h ) 1) is expected for the isotherms measured in a very low range of A ion concentration, whereas for extremely high values of CA (or for extremely low values of CB, eventually) the value of 1/h drops to 1/R which corresponds to a strong or very strong surface heterogeneity in the adsorption model. The applicability of our approach is illustrated by the analysis of four sets of experimental data selected from the literature. We present here the analysis of the experimental data published originally by Papageorgiou et al. (30) in 2006 (reporting on biosorption of Cu2+, Cd2+, and Pb2+ ions by calcium alginate beads extracted from Laminaria digitata) and by Boucher et al. (31) in 2008 (reporting on biosorption VOL. 43, NO. 19, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 4. Values of the Best-Fit Parameters Obtained while Fitting the Data Presented in Figures 3 and 4 by Equation Set 19a Papageorgiou et al. (30) parameter values pH
Cu2+
parameter/units
Cd2+ -3
Boucher et al. (31) Pb2+
-3
pH
parameter values
-3
2.5
Ns [mol/g] κ [g/L] r2
2.95 · 10 13.7 0.990
2.65 · 10 5.94 0.994
3.35 · 10 233 0.997
3
2.29 · 10-4 0.710 0.920
3.5
Ns [mol/g] κ [g/L] r2
3.30 · 10-3 10.1 0.981
2.83 · 10-3 5.92 0.986
3.55 · 10-3 267 0.981
4
7.96 · 10-4 0.100 0.987
4.5
Ns [mol/g] κ [g/L] r2
3.32 · 10-3 13.1 0.992
2.87 · 10-3 5.94 0.985
3.64 · 10-3 193 0.998
5
9.41 · 10-4 0.107 0.956
pKa
2.48
2.58
2.78
3.78
a
The Ns values were estimated by the authors of the original papers by using the concept of adsorption capacity predicted by the Langmuir model for the data measured at the highest pH. They are equal to 2.76 × 10-3 mol/g (for Cu2+), 2.32 × 10-3 mol/g (for Cd2+), and 3.59 × 10-3 mol/g (for Pb2+) in the case of the data by Papageorgiou et al. (30) and 1.50 × 10-3 mol/g in the case of the data by Boucher et al. (31). The previously assumed stoichiometry (1:2) was accepted to calculate the above Ns values.
of Cu2+ by the raw biosorbent rapeseed husks). All details related to the experimental procedures can be found in the original papers. First, let us note that in both cases the mechanism of bivalent metal biosorption by the ion exchange (which involves mainly the carboxyl groups 30–32) can be accepted. The most probable stoichiometry for sorption of bivalent metal ions can be represented by R ) 2, β ) 1. The model based on these two assumptions was further extended by taking into account the acid-base properties of the biosorbent surface. Let us write the appropriate equation set:
{
Ns ) 2qA + q0 + qH κ)
qA10-2pH 2 qH CA
(19)
q010-pH Ka ) qH
in which q0 denotes the concentration of free binding sites (in [mol/g]). The q0 and qH parameters can be eliminated from the above equation set, thus, the analysis of data by using this set requires adjusting only three parameters: Ns, Ka, and κ. The fitting procedure was performed for twelve data sets (the sorption isotherms characteristic of each system were recorded at three different pH values). Other details are described in the SI. The results of applying equation set 19 to analyze the sorption isotherms are presented in Figures 3 and 4. The values of the determined best-fit parameters are collected in Table 4. The proposed model offers reasonably well fitting of data, providing approximately constant values of the best-fit parameters for a given system. Moreover, as supposed, the Ns and pKa parameters have similar values for all three systems containing the alginate as the biosorbent (i.e., 2.48, 2.58, and 2.78 for Cu2+, Cd2+, and Pb2+, respectively). The observed differences between the Ns values obtained for different systems can be explained by deviations from the assumed stoichiometry 1:2 and by contribution of mechanisms different from ion exchange (electrostatic interactions, for instance). This can also be the reason why the estimated pKa values are slightly lower than their typical values reported in literature (in the case of alginate they vary from 3 to 5). The footnote of Table 4 contains the Ns values estimated by 7470
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the authors of the original data. The significant difference between the Ns values characterizing both sorbents is reflected by the maximum uptake values obtained by the authors of both papers (for the Cu2+/alginate system it is equal to 87.39 mg/g while for the Cu2+/rapeseed husks system it is only 36.6 mg/g). In the case of the Cu2+/rapeseed husks system one can observed significant differences between the values of k and Ns parameters obtained for pH ) 3 and those characteristic of higher pHs. A possible explanation may be different mechanism of biosorption dominating at lower pH values. On the other hand, there are only four data points measured at pH ) 3 and the monitored qA values are extremely low. This may suggest a significant influence of the random experimental error. Nevertheless, the adjusted value of pKa is in agreement with those suggested by Boucher et al. (31), i.e., it is between 3 and 4. (The FTIR investigation of functional groups present on the surface of rapeseed husks confirmed that the binding of Cu is due to some -OH groups, either in hydroxyl or carboxyl groups. However, the pH-dependent binding behavior suggests that carboxyl groups are responsible for the binding of Cu to a greater extent than other groups when considering the studied range of pH (31)). Further, the obtained Ns values are slightly lower than those reported by Boucher et al. (see Table 4); this suggests that not only bidentate but also monodentate surface complexes may be formed during biosorption process. The results presented here show that even such a simplified model is able to predict accurately the behavior of different biosorption systems and that the surface heterogeneity claimed on the basis of adsorption model may be only “apparent” heterogeneity. For the four systems investigated here these heterogeneity effects can be satisfactorily explained on the grounds of ionic exchange model represented by equation set 19. Finally, it is obvious that in the real (physical) biosorption systems all the mentioned above effects can be present simultaneously, i.e., the effect of stoichiometry can combine with the IECD functions in a much more complicated way than the two simple models presented here. Further studies on this subject are needed.
Supporting Information Available Principles of the CA approach and the method of deriving eq 9; ion exchange isotherms: basic mathematical principles;
expressing the heterogeneity in terms of discrete distribution of the ion exchange constant; descriptions of the procedure of comparative analysis of models and the data fitting procedure; more detailed results of model calculations; Figure S1, application of the CA method for the three different stoichiometries (1:1, 1:2 and 1:3); Figure S2, a symbolic illustration of the procedure of the comparative analysis of models; Figures S3 and S4, graphical representation of the selected results of our numerical calculations. This material is available free of charge via the Internet at http:// pubs.acs.org.
Acknowledgments W.P. acknowledges the financial support of the Foundation for Polish Science (START program, 2009).
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