An Analytical Isotherm Equation (CONICA) for Nonideal Mono- and Bidentate Competitive Ion Adsorption to Heterogeneous Surfaces

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

183, 35–50 (1996)

0516

An Analytical Isotherm Equation (CONICA) for Nonideal Mono- and Bidentate Competitive Ion Adsorption to Heterogeneous Surfaces WILLEM H.

RIEMSDIJK,* ,1 JOHANNES C. M. DE WIT,† SIPKO L. J. MOUS,* LUUK K. KOOPAL,‡ AND DAVID G. KINNIBURGH * ,2

VAN

*Department of Soil Science and Plant Nutrition, Wageningen Agricultural University, P.O. Box 8005, 6700 EC Wageningen, The Netherlands; †Tauw Milieu, P.O. Box 133, 7400 AC Deventer, The Netherlands; and ‡Department of Physical and Colloid Chemistry, Wageningen Agricultural University, P.O. Box 8038, 6700 EK, The Netherlands Received November 20, 1995; accepted May 29, 1996

dominantly of a ‘‘carboxylic’’ and ‘‘phenolic’’ nature (1), but as with the oxides differences in the chemical environment of the two types of groups lead to a wide range of affinity constants, i.e., a large chemical heterogeneity. This is reflected in the distribution of proton affinities of these materials (2). For humic acids, the proton affinity distribution shows two broad peaks with approximately the same number of sites per peak. Calculation of the local electrostatic potential (3–5) as a function of pH shows that the electrostatic effects are much less pronounced than for metal oxides (6) but are nevertheless significant. A characteristic of metal ion binding to metal oxides and humic and fulvic acids is that at constant pH and salt concentration, a Freundlich (log–log) plot shows that there is a slope of less than one over a fairly wide range of free metal ion concentrations. In general, the slope is characteristic of both the metal ion and the sorbent. For humic acids, the initial slope of such Freundlich plots does not seem to approach the Henry limit of one even at extremely low free metal ion concentrations (7), whereas for metal oxides, it does so at very low metal loadings (8). At high metal loadings, the slopes shown by such plots tend to decrease. The binding of protons by humic substances can be described with a bimodal Langmuir–Freundlich (LF) model (5). This can be combined with a cylindrical double-layer model (5) or with a Donnan-type model (7) to take into account the electrostatic interactions and so can reflect the effect of changes in background electrolyte concentration. The pH-dependent binding of both calcium and cadmium have been described with a bimodal competitive LF model (9). However, the ‘‘fully coupled’’ case in which the affinity distribution of the metal ion has the same shape as that of the proton distribution but is simply shifted along the affinity axis fitted the data rather poorly. Only when the two individual monomodal distributions were allowed to shift independently was a reasonable fit obtained for cadmium and calcium binding. This approach completely fails to account for the pH-dependent binding of copper to the same humic acid.

Analytical isotherm equations for the competitive binding of protons and other cations to heterogeneous surfaces have been derived. These extend our earlier nonideal competitive adsorption (NICA) equation by considering bidentate as well as monodentate binding of ions. The protonation of these bidentate sites occurs in two COnsecutive steps. The CONICA model equations separate the apparent heterogeneity into a generic heterogeneity characteristic of all of the surface sites and an ion-specific contribution characteristic of each type of ion. The CONICA model was combined with a simple Donnan model to account for electrostatic effects and was fitted both to proton binding at various ionic strengths and to the pH-dependent binding of Cu (pH 4, 6, and 8) to a purified peat humic acid. Model fits were good. The measured H / /Cu 2/ exchange ratio is significantly greater than one for the whole pH range, which indicates the importance of the bidentate binding mechanism for copper. q 1996 Academic Press, Inc. Key Words: adsorption; heterogeneity; humic; isotherm; copper; proton binding.

INTRODUCTION

Natural sorbents such as the metal oxides (and hydroxides) and humic and fulvic acids have reactive oxygen-containing surface groups that can act as binding sites for protons and metal ions. This binding is influenced by electrostatic forces of a coulombic nature and by an intrinsic affinity of the adsorbing ion for particular binding sites. The variable charge, variable potential characteristics are particularly strong for metal oxides which show a near Nernstian response. Oxygen at the surface of metal oxides can be singly, doubly, or triply coordinated to the underlying metal atoms of the solid. This makes the proton affinity of the various reactive groups quite different and leads to chemical heterogeneity at the molecular scale. The reactive oxygens of humic and fulvic acids are pre1 2

To whom correspondence should be addressed. On leave from the British Geological Survey, Wallingford, Oxon, UK. 35

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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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FIG. 1. The various configurations of phthalate- and salicylate-type functional groups that may be involved in metal binding by humic substances. This example shows copper as the bound metal ion ( M Å Cu).

These observations have led to the development of the NonIdeal Competitive Adsorption (NICA) equation which is based on a nonideal competitive Henderson–Hasselbalch/ Rudzinski local isotherm. The NICA equation separates effects arising from ion-specific nonideality or ion-specific heterogeneity from those relating to a generic heterogeneity that is a surface characteristic (10). This generic heterogeneity is assumed to be described by a LF (Sips) distribution (11). Since humic acids clearly show a bimodal distribution of proton affinities, the bimodal form of the NICA equation is appropriate if a broad description of their behavior is required. Such an approach has given very good results for the description of both proton binding and the pH-dependent binding of calcium, cadmium, and copper to a humic acid sample (12). The results suggested that the relative binding to the ‘‘phenolic’’ (high proton affinity) sites was smallest for calcium and greatest for copper. The measured proton/ metal ion exchange ratio for calcium and cadmium binding by humic acid is less than one and is approximately reproduced by the bimodal NICA model (7, 12). However, the measured exchange ratio for copper was found to be between one and two and varied with the amount of copper bound. This behavior cannot be reproduced with the bimodal NICA model since an underlying assumption of the model is that of monodentate binding and this inevitably leads to a maximum proton/metal ion exchange ratio of one. The binding that the NICA approach ascribes to ‘‘phenolic’’-type groups on the basis of an assumed 1:1 stoichiometry may in reality be due to the formation of bidentate structures. The NICA approach therefore has a weakness if a complete description of copper binding to humics is required (as do other approaches). A likely shortcoming is that in the case of Cu, bidentate chelate-type structures can form as a result of binding with salicylate- and phthalatetype moieties (13, 14) (Fig. 1). Spectroscopic evidence of Cu binding to humic substances is generally consistent with such configurations although it is usually not unambiguous (1). Bidentate binding is not accounted for in the NICA

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model and its inclusion would in principle allow exchange ratios greater than one. The aim of this paper is to derive NICA-type equations that can be used to describe binding to heterogeneous, nonideal systems in which both monodentate and bidentate binding can occur. The new model is called the CONICA model since it has strong similarities with the NICA model and considers proton binding as a two-step COnsecutive reaction. The major part of the mathematical derivation of the equations with discussion will be given in the Appendix. The derivation of the two-step protonation reaction will be given and discussed in the main text. In the Appendix the derivation is first done for a combination of proton binding and monodentate metal ion binding to the low affinity (‘‘carboxylic’’) groups, then proton binding and bidentate metal ion binding, and finally the full model which considers proton binding, monodentate, and bidentate metal ion binding simultaneously. For the sake of simplicity we will omit the charges of the reactive sites and the ions in the equations. The actual charges become important when the equations are combined with an electrostatic model. The applicability of the model will be tested in combination with a simple Donnan model to account for electrostatic interactions using data on pH-dependent binding of Cu to a humic acid. CONSECUTIVE NONIDEAL COMPETITIVE ADSORPTION

The first attempt to obtain analytical expressions for ion binding to heterogeneous surfaces based on a two-step protonation reaction was made by van Riemsdijk et al. (15). It was assumed that the binding constant of the first proton, KSH , is a distributed parameter composed of two contributions, KSH Å KL,1kSH , where KL,1 is the part that is subject to the generic heterogeneity and is independent of the adsorbing ion and kSH is the ion-specific part that is not influenced by the heterogeneity. The second binding constant, KSH2 , was assumed to be effectively constant. The distribution of the product KSH 1 KSH2 then has the same shape as that of KSH but is shifted along the affinity axis. Although such an assumption is convenient for the derivation of analytical competitive binding equations, it may not be physically very realistic and was later relaxed (16). The CONICA model assumes that KSH2 , like KSH , is a distributed parameter made up of two independent contributions, KSH2 Å KL,2kSH2 . In this new model, the parameter KSH represents the affinity of the high-affinity ‘‘phenolic’’ sites and KSH2 represents the affinity of the ‘‘carboxylic’’ sites, which have a lower proton affinity also. We further make the assumption that the generic heterogeneity for the first protonation step is independent of the generic heterogeneity for the second protonation step; i.e., the distribution functions for KL,1 and KL,2 are independent. It is implicitly assumed that the number of

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CONICA MODEL

‘‘carboxylic’’- and ‘‘phenolic’’-type groups are present in similar numbers. At least for humic acids, this assumption does not seem too far from reality. We discuss later why this assumption is not as critical as it first might appear.

As in the NICA model (10), we use the competitive Henderson–Hasselbalch isotherm as the nonideal local isotherm to account for the nonideality and ion-specific heterogeneity, UL,i Å

Two-Step Protonation, S–SH–SH2 In order to describe bidentate binding in combination with monodentate binding it is assumed that a reactive unit (‘‘site’’) can be protonated in two consecutive steps: S / H T SH SH / H T SH2

KSH

[1]

KSH2 .

[2]

Each site is therefore viewed as having the potential for binding up to two protons: uH S vH

first proton, high affinity (‘‘phenolic’’) second proton, low affinity (‘‘carboxylic’’).

A similar stepwise protonation scheme is also commonly used for describing the charging of variable charge metal oxide surfaces (the ‘‘2 pK’’ model). First let us consider the way that the distribution of KL,2 influences the formation of the SH and SH2 species for sites which have a proton affinity KSH Å KL,1kSH for the first proton. For the sites with an ion affinity proportional to KL,1 and KL,2 , we can define the partial fractions U *L,SH (KL,1 , KL,2 ) and U *L,SH2 (KL,1 , KL,2 ), U *L,SH (KL,1 , KL,2 ) å U *L,SH2 (KL,1 , KL,2 ) å

[SH]KL,1,KL,2

[3]

[SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2 [SH2 ]KL,1,KL,2 [SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2

The parameters ki and ni are ion specific whereas KL is a distributed parameter that is a surface characteristic and is thus assumed to be the same for all adsorbing ions. The variable ci can be either an ion concentration or activity. The exponent ni accounts for the ion-specific nonideality. Equation [7] simplifies to the competitive Langmuir equation when all nonideality exponents are equal to one (i.e., ideality). Essentially the same equation has been derived by Rudzinski et al. (17, 18), assuming a heterogeneous surface where there is no correlation between the adsorption energies of the different adsorbing ions. We therefore refer to the local isotherm as the Henderson–Hasselbalch/Rudzinski equation. This isotherm is equivalent to assuming that all local nonideality is ion specific. Using this local isotherm, the concentration of [SH2 ]KL,1,KL,2 can be expressed as [SH2 ]KL,1,KL,2 Å KL,2 (kSH2 H) n2 [SH]KL,1,KL,2 ,

U *T,SH (KL,1 ) Å

*1/K

1 f (log KL,2 )d(log KL,2 ) n2 L,2 (kSH2 H)

[4] Å

*1/K

KL,2 (kSH2 H) n2 L,2

(kSH2 H) n2

(KL,1 , KL,2 ) f (log KL,2 )d(log KL,2 )

[5]

U *T,SH2 Å

U *T,SH2 (KL,1 ) Å

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f (log KL,2 )d(log KL,2 ).

[10]

An analytical solution of this equation can be obtained when we use a Langmuir–Freundlich type affinity distribution (quasi-Gaussian or Sips (11)) for the distribution of KL,2 . We then get U *T,SH Å

L,SH

[8]

in which H is the concentration or activity of protons and n2 accounts for the nonideal behavior of the second proton. We can now rewrite Eqs. [5] and [6] as

U *T,SH (KL,1 )

* U*

[7]

U *T,SH2 (KL,1 )

,

where the square brackets indicate species concentrations (in some convenient units such as mol kg 01 ) and the subscript KL,1 , KL,2 indicates that only sites with a certain affinity KL,1 and KL,2 are considered. The total partial fractions of the SH and SH2 species, U *T,SH (KL,1 ) and U *T,SH2 (KL,1 ), with respect to all sites with an affinity KL,1 are then given by

Å

KL (ki ci ) ni . 1 / SKL (ki ci ) ni

1

[11]

1 / (KH SH2 H) n2 p2 (KH SH2 H) n2 p2 1 / (KH SH2 H) n2 p2

,

[12]

where p2 is the width of the Sips distribution of KL,2 . Note that we can write U *T,SH and U *T,SH2 instead of U *T,SH (KL,1 )

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and U *T,SH2 (KL,1 ), respectively, because these ‘‘total’’ partial fractions do not depend on KL,1 . A similar relation to that given in Eq. [8] can be derived from Eq. [12] for all sites characterized by the same KL,1 and independent of KL,2 : [SH2 ]KL,1 Å (KH SH2 H) n2 p2 [SH]KL,1 .

[13]

This relation reflects the assumption that the second protonation step does not depend on the shape of the distribution function for KL,1 yet is dependent on the prior binding of a proton on that site. Using this relation we can derive expressions for the fraction of sites with an affinity KL,1 which have adsorbed one or two protons: [SH]KL,1

UL,SH (KL,1 ) Å

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1 KL,1 (kSH H) n1 (KH SH2 H) n2 p2

1 / KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2

.

[15]

Note that these relations depend on the distribution of KL,1 only. The total binding of SH and SH2 is obtained by integrating these ‘‘local’’ fractions over KL,1 :

*U Å*U

L,SH

(KL,1 ) f (log KL,1 )d(log KL,1 )

[16]

(KL,1 ) f (log KL,1 )d(log KL,1 ).

[17]

L,SH2

The solution of these equations is easily found if we first multiply the numerator and denominator of the kernel by [SH]KL,1 / [SH2 ]KL,1 , UT,SH Å U *T,SH

* (U

L,SH

(KL,1 )

/ UL,SH2 (KL,1 )) f (log KL,1 )d(log KL,1 )

UT,SH2 Å U *T,SH2

* (U

L,SH

[18]

(KL,1 )

/ UL,SH2 (KL,1 )) f (log KL,1 )d(log KL,1 ).

[19]

In Eqs. [18] and [19], the total partial fractions are placed in front of the integral because these fractions do not depend on KL,1 . The total binding is thus obtained by multiplying the expression for the partial fraction by the fraction of sites which

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[(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 ] p1

1 / [(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 ] p1

[20]

(KH SH2 H) n2 p2

1 / (KH SH2 H) n2 p2

1

[14]

[SH2 ]KL,1

UT,SH Å

1

1

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1

UL,SH2 (KL,1 ) Å

UT,SH2

UT,SH Å

UT,SH2 Å

KL,1 (kSH H) n1 Å n1 1 / KL,1 (kSH H) / KL,1 (kSH H) n1 (KH SH2 H) n2 p2

Å

are not in the reference state, i.e., UT,SH / UT,SH2 (the ‘‘reference’’ state refers to the fully deprotonated species, S). If we also assume a Henderson–Hasselbalch/Rudzinski local adsorption equation for the first protonation step, the local fraction UL,SH / UL,SH2 becomes mathematically equivalent to a monocomponent Langmuir relation. For a Sips distribution with p1 as the width of the generic distribution, the solutions of Eqs. [18] and [19] are given by

[(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 ] p1

1 / [(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 ] p1

.

[21]

Equations [ 20 ] and [ 21] describe the fractional coverage of the surface with the two protonated surface species. In order to convert this fractional coverage to an absolute coverage, it is multiplied by the site density, Ns . Some simulations based on Eqs. [ 20 ] and [ 21] are shown in Fig. 2. These are for a hypothetical material with 5.0 mol kg 01 of sites which has two reasonably well-separated ˜ SH Å 8 and log KH SH2 Å 4. modal proton affinities of log K In the homogeneous ideal case ( p1 Å p2 Å 1 and n1 Å n2 Å 1 ) , the inflection points in the proton binding curve ˜ values occur at positions given by the respective log K / ( i.e., at log [ H ] Å 08 and 04 ) with a very distinct plateau midway between at log [ H / ] Å 06 ( Fig. 2a ) . The homogeneous ideal case is identical to a two-site competitive Langmuir isotherm. In contrast, with the heterogeneous ideal case ( p1 Å p2 Å 0.5 ) , the proton binding increases almost linearly between log [ H / ] of 09 and 03. ˜ ’s is distributed ( Fig. 2b ) , the When only one of the log K corresponding part of the proton binding curve becomes smoothed while the undistributed part retains a distinct inflection. The case in which only the high-affinity proton is distributed ( p1 Å 0.5 with p2 Å 1 ) corresponds with the heterogeneous 2-p K model for the protonation of oxides that was considered earlier ( 15 ) . The total fraction of sites not in the reference state, UT,SH / UT,SH2 , is equal to the second term of the right-hand side of Eqs. [20] and [21]. At low pH, the fraction of sites in the reference state becomes negligible. Under these conditions the fraction UT,SH / UT,SH2 is approximately equal to one. Equation [21] then simplifies to a NICA equation or even the Langmuir–Freundlich equation at low pH. At high pH the fraction UT,SH2 becomes negligible, which is the case

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FIG. 2. Simulated proton binding according to the CONICA model for both ideal homogeneous and ideal heterogeneous cases: (a) p1 Å p2 Å 1 and p1 Å p2 Å 0.5, and (b) p1 Å 1, p2 Å 0.5, and p1 Å 0.5, p2 Å 1, both with n1 Å n2 Å 1. Other isotherm parameters are indicated on the figures.

when (KH SH2 H) n2p2 ! 1. Under these conditions, Eq. [20] simplifies to

UT,T Å UT,SH / UT,SH2 / SUT,SHMi / SUT,SMi Å

UT,SH É

(KH SH H) n1p1 . 1 / (KH SH H) n1p1

Å

Two-Step Protonation, Monodentate and Bidentate Metal Ion Binding, S–SH–SH2 –SHM–SM In the Appendix the mathematical derivation of the twostep protonation in combination with monodentate and bidentate metal ion binding is given. It is shown that the binding of the individual species are obtained by taking fractions of fractions. This concept is illustrated in Fig. 3 for proton binding in the presence of mono- and bidentate metal ion binding. First the fraction UT,T , i.e., the fraction of sites that are not in the reference state, S is calculated:

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L,SH

/ UL,SH2 / SUL,SHMi / SUL,SMi )

1 f (log KL,1 )d(log KL,1 )

[22]

Equation [22] is again of the NICA or Langmuir–Freundlich type. Initial estimates of the values of KH SH2 and n2 p2 and ˜ SH and n1 p1 can therefore be obtained from the positions K and widths of the peaks in the proton affinity distribution. The CONICA equations, Eqs. [20] and [21], simplify to the equations that were derived earlier (15) when it is assumed that the second proton step is nondistributed and when any nonideality is ignored, i.e., if p2 Å 1 and p1 Å m and all ni are set equal to 1. The differences between the CONICA approach and the bimodal NICA approach for protonation are less than might appear from the large differences in the expressions, provided that the two peaks of the affinity distribution are well separated as in the example discussed. Note that in the bimodal NICA approach the indices for the parameters for the carboxylic and phenolic peaks are the reverse of those used here, i.e., the generic distribution KL,1 in the bimodal NICA approach is used to refer to the ‘‘carboxylic’’-type sites (12).

* (U

[(KH SH H) n1 / (KH SH H) n1 A p2 / B] p1 . 1 / [(KH SH H) n1 / (KH SH H) n1 A p2 / B] p1

[23]

with A Å [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] B Å S(KH SMi Mi ) £i ]. This fraction can then be subdivided into two groups. These are the sites that have bound a metal ion by bidentate binding and the sites that have bound at least one proton. The group that contains sites that have bound at least one proton is given by the sum fraction ( UT,SH / UT,SH2 / ( UT,SHMi ): UT,SH / UT,SH2 / SUT,SHMi Å

* (U

Å

(KH SH H) n1 / (KH SH H) n1 Ap2 rUT,T . (KH SH H) n1 / (KH SH H) n1 A p2 / B

L,SH

/ UL,SH2 / SUL,SHMi ) f (log KL,1 )d(log KL,1 )

[24]

This group can be further subdivided into the fraction of sites which bind only one proton and the fraction of sites that are doubly occupied. This last group can finally be subdivided into the sites that have bound two protons and the sites that have bound one proton and one metal ion, i.e., the proton and the metal ion compete for the low affinity site. This subdividing leads to analytical expressions for the total binding of the individual species:

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FIG. 3. Figure showing the derivation of the various partial fractions on which the CONICA equations are based. Numbers in square brackets refer to equation numbers in the text.

UT,SH Å

1 / [(KH SH2 H)

n2

1 / S(KH SHMi Mi ) li ] p2

1 ( UT,SH / UT,SH2 / SUT,SHMi )

UT,SH2 Å

(KH SH2 H) n2

(KH SH2 H) n2 / S(KH SHMi Mi ) li 1

[(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 1 ( UT,SH / UT,SH2 / SUT,SHMi )

UT,SHMj Å

[26]

(KH SHMj Mj ) l j

(KH SH2 H) n2 / S(KH SHMi Mi ) li 1

[(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 1 ( UT,SH / UT,SH2 / SUT,SHMi )

UT,SMj Å

[25]

(KH SMj Mj ) £j

rUT,T . (KH SH H) n1 / (KH SH H) n1 A p2 / B

[27] [28]

A summary of the various combinations of proton and metal ion binding considered by Eqs. [ 25 ] to [ 28 ] is given in Table 1. These equations are comprehensive in the sense that under the appropriate conditions they reduce to the simpler cases derived earlier, e.g., proton binding in the absence of metal, and bidentate metal ion binding without monodentate binding. In the appendix it is shown that the comprehensive equations reduce under certain conditions to a monomodal NICA equation, or even a Freundlich equation. In situations in which both monodentate and bidentate

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metal ion binding occur, the overall extent of metal ion binding and its response to changes in pH will be strongly dependent on the actual values of the various isotherm parameters. The deprotonation of the high-affinity proton site at high pH will lower the probability of the formation of the SHM species and in principle could lead to a decrease in the total amount of metal bound ( especially ˜ SM õ K ˜ SHM ) . However, such a decrease when li £ £i and K is not usually observed and so in practice the various combinations of parameters observed is somewhat constrained. The bidentate SM species will tend to become important at high pH and will lead to an overall increase in the total metal ion binding with increasing pH. A possible decrease in metal ion binding is also less likely at high pH if the CONICA model is combined with an explicit electrostatic model since the increase in negative charge will increase the concentration of positively charged counterions which may contribute significantly to the total metal ion binding at high pH. This effect has been shown to be important for the pH-dependent binding of calcium to humic acid ( 7 ) . Although the CONICA model implicitly assumes that the maximum number of high- and low-affinity protons that can be bound is similar, in practice this need not necessarily be so. The various exponents that are always present in the CONICA equations have a strong influence on the actual amount of proton ( and metal ion ) binding and can result in an upper limit to the effective occupancy of certain ‘‘low affinity’’ sites. These sites may simply never be occupied under a certain range of experimental conditions. Similar arguments apply to the number of proton binding sites and the maximum number of metal ions bound. The flexibility of the model is therefore considerable.

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TABLE 1 The Combinations of Proton and Metal Binding to the High and Low Proton Affinity Locations That Are and Are Not Considered in the CONICA Equations, Eqs. [23] to [28] Combinations considered and species Combinations not considered Location High proton affinity location (1) Low proton affinity location (2)

S

SH

SH2

SHM

SM

—

H

H

H

M

—

—

M

M

—

—

H

M

—

H

M

H

M

Note. —, unoccupied; H, occupied by a proton; M, occupied by a metal ion. Since the high proton affinity location is in principle the first to be protonated, it is referred to as location 1 with parameters, p1 , n1 , etc. The low affinity location is referred to by the subscript 2. Although in the SM species, the metal ion is likely to be shared equally between the high and low affinity locations, mathematically it is treated as binding to the high affinity location (and so inherits other characteristics of that location).

APPLICATION OF THE CONICA MODEL

Proton and Copper Binding to Humic Acid A thorough test of the CONICA model is outside the scope of this paper, but we demonstrate the ability of the model to describe the pH-dependent binding of Cu by a humic acid and also compare the calculated H / /Cu 2/ exchange ratios with the observed ones. As we have done previously for the NICA model, we combine the CONICA model with the Donnan model to take into account the electrostatic interactions (7, 19). The Donnan model enables ‘‘surface’’ concentrations to be used in the CONICA equations instead of bulk solution concentrations and it also allows for the binding of cations (and repulsion of anions) in the Donnan phase itself. The PPHA material and the methods of data collection have been described in detail elsewhere (5, 12). The acid– base behavior of the redispersed PPHA was determined by a series of acid–base titrations at various ionic strengths (0.002 to 0.3 M KNO3 ) over the pH range from pH 3.5 to 10.5. The Cu binding experiments were performed at approximately pH 4, 6, and 8 in a 0.1 M KNO3 background electrolyte. The bulk solution concentration variables are all expressed in terms of the concentration of free ions. As before, the parameter estimation was conveniently performed in two stages, first, by fitting the acid–base data, and second, using the parameters derived from the first step, by fitting the metal binding data (7). Analysis of the proton binding data (Step 1) enabled estimates of the following parameters to be made: the initial charge on the humic material, an empirical parameter describing the Donnan behavior (b), the number of sites (Ns ) and the proton affinity and apparent heterogeneity for the two protons, i.e., log KH SH1 and m1 , and log KH SH2 and m2 (m1 Å n1 1 p1 and m2 Å n2 1 p2 ). In theory, it should be possible to separate n1 from p1 (but not n2 from p2 ) using the proton data alone, Eqs. [20] and [21], but in practice the correlation between n1 and p1 is so

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large that only m1 can be estimated with precision. The separation of m1 into n1 and p1 is best achieved using the metal binding data just as it is for m2 . Analysis of these data without further adjustment of the parameters derived from the proton titration curves gave the Cu binding parameters. All fitting was performed by a nonlinear least-squares program. This program can be obtained from D.G.K. The resulting fits of the CONICA–Donnan model to both the proton charging curves (Fig. 4) and the Cu binding data at the three pH’s (Fig. 5) are excellent. The overall goodness-of-fit is similar to those of the NICA–Donnan model (7). As might be expected, many of the parameter values are also similar (Table 2). The CONICA model has one

FIG. 4. Measured and calculated charging curves for the purified peat humic acid (PPHA) at four different ionic strengths according to the CONICA–Donnan model. The values of the fitted parameters are given in Table 2. The ionic strengths indicated are approximate. The ionic strengths varied somewhat between duplicate experiments, especially at high pH; the calculations used the exact ionic strength.

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does the SHCu species become dominant. The amount of Cu 2/ bound in the Donnan phase never exceeds 16% of the total Cu bound. It is the greatest percentage at low pH where Cu binding is smallest. Exchange Ratios

FIG. 5. Measured and calculated copper binding curves to the PPHA at pH 4, 6, and 8 according to the CONICA–Donnan model. The values of the fitted parameters are given in Table 2.

less parameter than the bimodal NICA model, since instead of the number of sites in the ‘‘carboxylic’’ and ‘‘phenolic’’ distributions being completely independent, as in the NICA model, the CONICA model defines the number of bidentate sites, each of which is able to bind up to two protons. Analysis of the species involved in the Cu binding (Fig. 6) indicate that at low Cu 2/ concentrations (log[Cu 2/ ] õ 06), the bidentate species, SCu, is the dominant species at all pH’s. Only at higher Cu 2/ concentrations and lower pH

The discussion above has shown that the behavior of the CONICA model is in many respects similar to that of the (bimodal) NICA model. The main reason for deriving the new equations has been to extend the possibility of using the NICA-type isotherms for describing bidentate binding. Bidentate binding cannot be properly simulated with the standard NICA model since a basic assumption of this is that all binding is monodentate. Therefore, aside from electrostatic effects, the maximum number of protons that can be released per metal ion bound is one. Experimental data for Cu binding to humic acids indicate that the exchange ratio is usually between 1 and 2 (12, 20). Gamble et al. (13) suggested that salicylate-type and phthalate-type structures might be responsible for Cu binding by humic acid and subsequent spectroscopic studies are consistent with a bidentate Cu–humic acid structure (21). Reaction of metal ions with bidentate sites can in principle release up to two protons. Electrostatic interactions can also affect the exchange ratio. For example, we have shown that by using a simple Donnan model in conjunction with the NICA model, the calculated exchange ratios for Cu binding to a purified peat humic acid (PPHA) can exceed one but only just (7). This increase in the exchange ratio arises primarily from the effect

TABLE 2 Optimized Parameter Values for the CONICA–Donnan and NICA–Donnan Models for Proton and Copper Binding to a Purified Peat Pumic Acid Corresponding parametera

Parameter value

CONICA–Donnan

NICA–Donnan

CONICA–Donnan

NICA–Donnan

Ns (mol kg01) b (Donnan) log KH SH2 n2 p2 ˜ SH log K n1 p1 ˜ SHCu log K lCu ˜ SCu log K nCu

(Qmax1 / Qmax2)/2 (mol kg01) b (Donnan) ˜ H,1 log K nH,1 p1 ˜ H,2 log K nH,2 p2 ˜ Cu,1 log K nCu,1 ˜ Cu,2 log K nCu,2

2.70 0.323 2.85 0.82 0.58 8.00 0.51 0.71 0.12 0.52 5.21 0.33

(2.74 / 3.54)/2 Å 3.14 0.43 2.98 0.86 0.54 8.73 0.57 0.54 0.40 0.52 6.42 0.32

a Most of the CONICA parameters are defined in Eqs. [23]–[28]. Ns is the site density in the CONICA model. The NICA–Donnan parameters and notation are from Kinniburgh et al. (7) and are based on an analysis of the same data set. Note that log KH SH2 refers to the second, weaker bound proton ˜ H,1 for the ‘‘carboxylic’’ peak in the NICA model; hence the switch between p1 and p2 , etc. between the two models. The and corresponds to log K Donnan b parameter is given by the empirical relation: log10 VD Å b(1 0 log10 I) 0 1, where VD is the Donnan volume in liter kg01 and I is the ionic strength. This relation is based on data for systems dominated by monovalent ions and was derived from Fig. 4 of Benedetti et al. (12).

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FIG. 6. Speciation of the Cu bound to the PPHA at various pH according to the CONICA–Donnan model. The percentage of the bound Cu is shown for (a) the SHCu species; (b) the SCu species, and (c) the Donnan-bound copper.

of the Cu binding on reducing the negative electrostatic potential close to the binding sites, which in turn leads to a reduction in the strength of the binding of all cations, including that of protons. This can lead to an additional release of protons and an increase in the H / /Cu 2/ exchange ratio. The H / /Cu 2/ exchange ratios were also measured during the Cu binding experiments using a pH stat approach and they have been calculated from the CONICA–Donnan and NICA–Donnan models. A comparison between the observed and calculated exchange ratios shows that the CONICA– Donnan model leads to a large improvement at pH 4, some improvements at pH 6, and no improvement at pH 8 (Fig. 7). The high measured exchange ratio at pH 8 is striking and can give insight into the binding mechanism. At pH 8,

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most of the protons are already desorbed from the ‘‘carboxylic’’ groups before any copper has been added. Monodentate or bidentate binding with carboxylic-type groups at pH 8 will therefore result in an exchange ratio that is much smaller than one. This mechanism can thus not explain the data. Bidentate binding with one weak acid group and one strong acid group will result in an exchange ratio of around one, as is observed in the application of the CONICA model. The formation of a bidentate complex with two ‘‘phenolic’’type groups would lead to an exchange ratio considerably higher than one at pH 8 and could thus in principle explain the data. However, this mechanism is not very likely considering the high amount of copper bound at pH 8. Another possibility is the binding of CuOH / species. Hydrolysis of

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VAN RIEMSDIJK ET AL.

FIG. 7. Measured and calculated H / /Cu 2/ exchange ratios, rex , as a function of function of free metal ion concentration and pH, (a) according to the NICA–Donnan model of Kinniburgh et al. (7) and (b) according to the CONICA–Donnan model presented here.

Cu 2/ would produce an extra proton in addition to those released by the competitive binding of Cu 2/ and H / . A similar fitting exercise with Ca 2/ and Cd 2/ binding data for the same humic acid has also shown that the CONICA model is able to provide a good fit to these isotherms at various pH’s as well as giving a reasonable description of the exchange ratios. Therefore it appears that the CONICA model has sufficient flexibility to be of general value in describing proton and metal ion binding by humic acid at least.

‘‘phenolic’’-type sites in the bimodal NICA approach can also be interpreted as being the result of the formation of bidentate complexes, as has been suggested earlier (12). The model can give a consistent description of proton binding at different salt levels and the pH-dependent binding of copper, cadmium, and calcium to the same humic acid.

CONCLUSIONS

In the system S–SH–SH2 –SHMj , we consider the reactions

Competitive ion binding to heterogeneous ligands is a complex phenomena. Analytical equations are useful because the competition between different metal ions for the same substrate can in principle be predicted once the model parameters have been derived from simpler systems, e.g., from systems containing a single metal ion. The exchange data for copper clearly indicate that bidentate binding is an important mechanism, whereas specific calcium binding is predominantly monodentate (7). A mechanistic metal ion binding model for humic acids should thus be able to deal with competitive monodentate and bidentate binding in combination with chemical heterogeneity. Although the CONICA equations have a more complex structure than the NICA equation, and are more difficult to derive, it is shown that they are fundamentally very similar. The number of parameters in the CONICA model is one less than in the corresponding bimodal NICA model. The CONICA model has the advantage that it explicitly takes into account a combination of monodentate and bidentate binding. The model analysis shows that the binding by the

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APPENDIX

Two-Step Protonation and Monodentate Binding, S–SH–SH2 –SHM

S / H S SH

KSH

[29]

SH / H S SH2

KSH2

[30]

KSHMj .

[31]

SH / Mj S SHMj

The metal ion is assumed to be in competition with the proton that binds to the low-affinity (‘‘carboxylic’’-type) sites. The metal binding is considered to be monodentate since the high-affinity proton sites are assumed to remain occupied. Binding of the metal ion to a high- or low-affinity site without a bound proton is not considered since it is likely that the formation of a bidentate SM site would be preferred. This point will be discussed in the next section. One could in theory also imagine monodentate binding to the high-affinity site with the low-affinity site still being protonated. This situation is not considered since it is assumed that when the metal ion is in competition with the high-affinity proton site it can also replace the weakly ad-

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CONICA MODEL

sorbed proton leading again to the bidentate option. In practice, the situation can be more complicated since steric factors may also affect the binding behavior. However, such steric effects are difficult to quantify for ill-defined reactive surfaces and complex molecules such as humic substances. These possible steric effects may be reflected in the value of the nonideality exponent. In deriving the expressions which describe the formation of SH, SH2 , and SHMi , we will start again by considering only those sites with an affinity KL,1 . The local competitive adsorption equations for a homogeneous subset of sites will again be assumed to be of the Henderson–Hasselbalch/Rudzinski type. It will be assumed that the generic Sips distribution for KL,2 is the same for both the SH2 and the SHMi complexes. In deriving expressions for the binding of the individual components it is easier to derive first an expression for the sum of the partial fractions of SH2 and of all monodentate metal ion complexes, since the sum of these partial fractions is mathematically equivalent to a monocomponent Langmuir equation,

U *L,SHMj (KL,1 , KL,2 )

U *L,T (KL,1 , KL,2 )

U *L,SH2 (KL,1 , KL,2 )

Å U *L,SH2 (KL,1 , KL,2 ) / SU *L,SHMi (KL,1 , KL,2 ) Å

Å

Å

Å

KL,2[(kSH2 H) n2 / S(kSHMi Mi ) li ] 1 / KL,2[(kSH2 H) n2 / S(kSHMi Mi ) li ]

[32]

U *T,T Å U *T,SH2 (KL,1 ) / SU *T,SHMi (KL,1 )

1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2

rU *L,T . [SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2

.

[33]

AID

[36]

[SHMi ]KL,1,KL,2 Å KL,2 (kSHMi Mi ) li [SH]KL,1,KL,2 ,

[37]

KL,2 (kSH2 H) n2 [SH]KL,1,KL,1 KL,2[(kSH2 H) n2 / S(KSHMi Mi ) li ][SH]KL,1,KL,1

rU *L,T [38]

Å

KL,2 (kSHMj Mj ) l j [SH]KL,1,KL,1 KL,2[(kSH2 H) n2 / S(kSHMi Mi ) li ][SH]KL,1,KL,2

rU *L,T . [39]

Note that KL,2 and [SH]KL,1,KL,2 cancel from the numerator and denominator of Eqs. [38] and [39]. These terms are therefore independent of the distribution of KL,2 and can be placed in front of the integration. If we further multiply the ˜ L,2 we may replace kSH2 and numerator and denominator by K kSHMi by KH SH2 and KH SHMi , respectively. In this way we find for the total partial fractions of the individual components

U *T,SH2 Å U *T,SHMj Å

(KH SH2 H) n2

rU *T,T (KH SH2 H) n2 / S(KH SHMi Mi ) li

(KH SHMj Mj ) l j

(KH SH2 H) n2 / S(KH SHMi Mi ) li

rU *T,T ,

[40]

[41]

where U *T,T is the sum of the ‘‘total’’ partial fractions, given by Eq. [33]. To derive the expressions for the total binding, we make use of the relation

U *L,SH2 (KL,1 , KL,2 )

Å

[SH2 ]KL,1,KL,2 Å KL,2 (kSH2 H) n2 [SH]KL,1,KL,2

we have

Note that we may write U *T,T instead of U *T,T (KL,1 ), since the right-hand side of Eq. [33] does not depend on KL,1 . Using the expression for the sum of the ‘‘local’’ partial fractions, given by Eq. [32], the local partial fractions for the individual components can be written as

Å

[35]

U *L,SHMj ( KL,1 , KL,2 )

,

[(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2

[SHMj ]KL,1,KL,1

and

Å

in which Mi is the concentration or activity of metal ion i and li is the nonideality factor. This equation can be integrated over KL,2 as in Eq. [6], resulting in

Å

[SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2

If we further make use of the relations

[SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2 [SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2

[SHMj ]KL,1,KL,2

[SH2 ]KL,1,KL,2 [SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2 [SH2 ]KL,1,KL,2

rU *L,T [SH2 ]KL,1,KL,2 / S[SHMi ]KL,1,KL,2

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[SH2 ]KL,1 / S[SHMi ]KL,1 [34]

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Å [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 [SH]KL,1 .

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[42]

46

VAN RIEMSDIJK ET AL.

This relation can be derived from the sum of the ‘‘total’’ partial fractions in the same way as Eq. [13] was derived from Eq. [12]. It is valid for all values of KL,1 and does not depend on the shape of KL,1 . Using this relation we can derive an expression for the sum of the SH2 and the monodentate metal ion complexes relative to the sites with a certain affinity KL,1 , which depends on the distribution of KL,1 only:

Å

[SH2 ]KL,1 / S[SHMi ]KL,1 [S]KL,1 / [SH]KL,1 / [SH2 ]KL,1 / S[SHMi ]KL,1 KL,1 (kSH H) n1 [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 ]

1 / KL,1[(kSH H) n1 / (kSH H) n1 [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 ]

UT,SH2 Å UT,SHMj Å

(KH SH2 H) n2

rU *T,T rUT,T

[48]

rU *T,T rUT,T , (KH SH2 H) n2 / S(KH SHMi Mi ) li

[49]

(KH SH2 H)

n2

/ S(KH SHMi Mi ) n2

(KH SHMj Mj ) l j

where U *T,T is given by Eq. [33] and UT,T is given by Eq. [45]. The fractional coverage with the SH2 species is therefore a fraction of a fraction of the number of sites not in the reference state. To complete the CONICA equations for the S–SH–SH2 –SHMj system, we can derive an expression for the total binding of SH in the same way as discussed previously for the case without metal ions:

UL,SH2 (KL,1 ) / SUL,SHMj (KL,1 ) Å

and integrating results in

.

UT,SH Å U *T,SH

[43]

* (U

L,SH

/ UL,SH2

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 )

After rearranging the kernel so that it again has the form of a monocomponent Langmuir equation, the expression for the sum of the SH2 and SHMi species is given by UT,SH2 / SUT,SHMi Å ( U *T,SH2 / SU *T,SHMi )

* (U

L,SH

/ UL,SH2

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 ).

[44]

The integral term in Eq. [44] is equal to the sum of all surface species which are not in the reference state, UT,T . Integrating this term gives [(KH SH H) n1 (1 / [(KH SH2 H) n2 UT,T Å

/ S(KH SHMi Mi ) li ] p2 )] p1

1 / [(KH SH H) n1 (1 / [(KH SH2 H) n2

.

L,SH

UT,SHMj Å U *T,SHMj

*

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Before we discuss the combined contributions of both monodentate and bidentate binding, we will first derive the expressions that are relevant if only bidentate complexes are formed. So, we will consider only the reactions

[46]

S / H S SH

K1

[51]

SH / H S SH2

K2

[52]

KSMj .

[53]

S / Mj S SMj ( UL,SH / UL,SH2

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 )

AID

[50]

Again, at low pH the number of sites in the reference state S is negligible. Under these conditions, the integral terms in Eqs. [46] and [47] are then approximately equal to 1 and Eqs. [48] and [49] reduce to essentially monodentate competitive NICA equations. The monodentate NICA equations result because at low pH, the high affinity sites will be fully protonated and thus do not affect the metal binding. A first estimate of the parameter lj can be obtained by considering competition of the proton with only one metal ion at a low free metal ion concentration. The metal term in Eq. [48] is then much smaller than the proton term and the equation reduces to a Freundlich isotherm in which the slope of a log–log plot gives lj . An initial estimate of p2 can then be found in the same way as suggested in Benedetti et al. (12).

/ UL,SH2

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 )

1 rUT,T . / S(KH SHMi Mi ) li ] p2

Two-Step Protonation with Bidentate Metal Ion Binding, S–SH–SH2 –SM

Using Eqs. [40] and [41], the fractional coverages UT,SH and UT,SHMj are obtained as

* (U

1 / [(KH SH2 H)

n2

[45]

/ S(KH SHMi Mi ) li ] p2 )] p1

UT,SH2 Å U *T,SH2

Å

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[47]

09-10-96 12:17:07

The metal ion is now in competition with both the high- and low-affinity protons. It will be assumed that the metal ion

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CONICA MODEL

competes primarily with the first (high-affinity) proton site, leading to the assumption that the shape of the generic affinity distribution for the first proton and the bidentate metal ion are identical. Departures from this idealized behavior can again be accounted for by the nonideality (or component-specific heterogeneity) exponent. Considering first again only sites with an affinity KL,1 and using the previously defined expressions for the partial fractions U *L,SH (KL,1 , KL,2 ) and U *L,SH2 (KL,1 , KL,2 ), U *L,SH (KL,1 , KL,2 ) Å U *L,SH2 (KL,1 , KL,2 ) Å

[SH]KL,1,KL,2

[54]

[SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2 [SH2 ]KL,1,KL,2

[SH]KL,1,KL,2 / [SH2 ]KL,1,KL,2

.

*

* (U

L,SH

Å

KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2 Å

[57]

[SH]KL,1

[60] In this way we have an expression which is again mathematically equivalent to a monocomponent Langmuir isotherm. Integrating this local fraction UL,T over KL,1 results in [(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2

1 / [(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

We then rewrite the kernel of the integral term as UL,SH / UL,SH2 Å

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1 n1

1 / KL,1 (kSH H) / KL,1 (kSH H) (KH SH2 H)

rUL,T [SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1

KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2

KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2

n2 p2

.

The factor KL,1 drops out of the numerator and denominator so that the first term can be placed in front of the integral. If we further multiply the numerator and denominator by ˜ SH and KH SMi , respec˜ L,1 we may replace kSH and kSMi by K K tively. After integration we then have UT,SH / UT,SH2 Å

/ SKL,1 (kSMi M) £i (KH SH2 H) n2 p2

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

/

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rUT,T .

/ S(KH SMi Mi ) £i

[59]

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rUL,T .

[62]

[SH2 ]KL,1 KL,1 (kSH H) n1 (KH SH2 H) n2 p2

[SH]KL,1 / [SH2 ]KL,1

n2 p2

UL,SH2 (KL,1 )

AID

.

/ SKL,1 (kSMi Mi ) £i

[58]

n1

/ S(KH SMi Mi ) £i ] p1 / S(KH SMi Mi ) £i ] p1

Å

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1

/ SKL,1 (kSMi M) (KH SH2 H)

Å

.

[61]

£i

Å

1 / KL,1 (kSH H) n1 / KL,1 (kSH H) n1 (KH SH2 H) n2 p2 / SKL,1 (kSMi Mi ) £i

UL,SH (KL,1 )

1 / KL,1 (kSH H) n1

/ SKL,1 (kSMi Mi ) £i

(K1 )

where the partial fractions U *T,SH and U *T,SH2 are identical to the expressions given in Eqs. [11] and [12]. The fractions UL,SH (KL,1 ) and UL,SH2 (KL,1 ) are given by

Å

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1

[56]

/ UL,SH2 (K1 )) f (log KL,1 )d(log KL,1 ),

Å

[SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1

UT,T Å

( UL,SH (K1 )

/ UL,SH2 (K1 )) f (log KL,1 )d(log KL,1 )

UT,SH2 Å U *T,SH2

UL,T Å UL,SH / UL,SH2 / SUL,SMi

[55]

The derivation of the binding of SH and SH2 is now straightforward. Following the derivation as described above in discussing the protonation behavior, we first integrate over KL,2 , yielding the partial fractions U *T,SH and U *T,SH2 relative to the sites with an affinity KL,1 . Using this result, the total binding of SH and SH2 species relative to all sites is then given by integration over KL,1 , UT,SH Å U *T,SH

The solution of the integral terms in Eqs. [56] and [57] can be easily obtained if we first derive an expression for the sum of the fractions of SH, SH2 , and SM:

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VAN RIEMSDIJK ET AL.

Substituting Eq. [63] into Eqs. [56] and [57], the total formation of the individual SH and SH2 species is equal to

The pH/pM shift for a given value of bound metal is then given by

UT,SH Å U *T,SH

D log[Mj ]

1

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 (KH SH H)

n1

n1

/ (KH SH H) (KH SH2 H)

rUT,T

n2 p2

/ S(KH SMi Mi ) £i

UT,SH2 Å U *T,SH2 1

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2 (KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

rUT,T .

[65]

/ S(KH SMi Mi ) £i

Note that U *T,SH , U *T,SH2 and UT,T are given by Eqs. [11], [12], and [61], respectively. In the same way as we have rewritten the sum of the fractions SH and SH2 , we can rewrite the kernel for the bidentate metal ion complexes, UT,SMj Å U *T,SMj

*

Å

[64]

1 D log[(KH SH H) n1 (1 / (KH SH2 H) n2 p2 )]. £j

The CONICA model with bidentate binding only simplifies to a NICA equation when the SH2 species are negligible compared to the SH species. For humic substances, this may occur at intermediate and high pH values (say above pH 7). Under these conditions (KH SH2 H) n2 p2 ! 1 and (KH SH H) n1 (KH SH2 H) n2 p2 ! (KH SH H) n1 and so Eq. [68] simplifies, as expected, to the NICA equation. Under these limiting conditions, the pH dependence of metal ion binding given by the CONICA equations with bidentate binding and of the basic NICA equation are therefore identical. Two–Step Protonation, Monodentate and Bidentate Metal Ion Binding, S–SH–SH2 –SHM–SM Combining both mono- and bidentate binding we get the system

( UL,SH (KL,1 ) / UL,SH2 (KL,1 )

/ SUL,SMi (KL,1 )) f (log KL,1 )d(log KL,1 )

Å

[SMj ]KL,1

[72]

SH / H S SH2

KSH2

[73]

S / Mj S SMj

KSMj

[74]

KSHMj .

[75]

SH / Mj S SHMj

(KH SMj Mj ) £ j

, [67]

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

and after integrating we have (KH SMj Mj ) £ j

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

rUT,T .

[68]

We make the same assumptions as in the previous systems; i.e., the generic heterogeneity related to the first step is the same for SH and SMj complexes and the generic heterogeneity for the second step is the same for SH2 and SHMj . Following the derivations of the previous systems, the total binding of SH, SH2 , SHMj , and SMj is given by UT,SH Å U *T,SH

* (U

The value of UT,T will be close to one at low pH. Equation [68] shows that at low pH and low metal ion binding a simple Freundlich-type of equation results, UT,SMj Å K*M j£ j ,

[69]

UT,SH2 Å U *T,SH2

L,SH

UT,SHMj Å U *T,SHMj

* (U

UT,SMj Å U *T,SMj

£

L,SH

(KH SH H) n1 / (KH SH H) n1 (KH SH2 H) n2 p2

/

6g16$$$$67

.

[70]

09-10-96 12:17:07

* (U

L,SH

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[77]

/ UL,SH2

[78]

/ UL,SH2 / SUL,SHMi

/ SUL,SMi ) f (log KL,1 )d(log KL,1 ),

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/ UL,SH2

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 ) j KH SM j

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* (U

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 )

where K* equals K* Å

/ UL,SH2

L,SH

/ SUL,SHMi ) f (log KL,1 )d(log KL,1 )

/ S(KH SMi Mi ) £i

AID

KSH

[SH]KL,1 / [SH2 ]KL,1 / S[SMi ]KL,1

/ S(KH SMi Mi ) £i

UT,SMj Å

S / H S SH

[66]

with U *T,SMj Å

[71]

[79]

49

CONICA MODEL

where the total partial fractions U *T,SH , U *T,SH2 , and U *T,SHMj are relative to the sites that have bound at least one proton and the total partial fraction U *T,SMj is relative to all sites that are not in the reference state S. The kernel of the integral term in Eq. [79] can be easily interpreted since the sum UL,T Å UL,SH / UL,SH2 / SUL,SHMi / SUL,SMi is mathematically equivalent to a monocomponent Langmuir isotherm equation. The result of this integration is

UT,SH / UT,SH2 / SUT,SHMi Å

* (U

* (U

L,SH

(KH SH H) n1 / (KH SH H) n1 [(KH SH2 H) n2 Å

(KH SH H) n1

/ UL,SH2 / SUL,SHMi / SUL,SMi )

[(KH SH H) n1 / (KH SH H) n1 [(KH SH2 H) n2

/ S(KH SHMi M) li ] p2 / S(KH SMi M) £i ] p1

1 / [(KH SH H) n1 / (KH SH H) n1 [(KH SH2 H) n2

.

[80]

/ S(KH SHMi Mi ) li ] p2 / S(KH SMi Mi ) £i ] p1

Using this result we can also easily integrate the kernel of the integral term in Eqs. [76], [77], and [78]. In the same way as before we rewrite the kernel as

1 1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 1 ( UT,SH / UT,SH2 / SUT,SHMi )

UT,SH2 Å UL,SH / UL,SH2 / SUL,SHM

[SH]KL,1 / [SH2 ]KL,1 / S[SHMi ]KL,1

(KH SH2 H) n2 / S(KH SHMi Mi ) li 1

[SH]KL,1 / [SH2 ]KL,1 / S[SHMi ]KL,1 [SH]KL,1 / [SH2 ]KL,1 / S[SHMi ]KL,1 / S[SMi ]KL,1 1

UT,SHMj Å

[SH]KL,1 / [SH2 ]KL,1 / S[SHMi ]KL,1 / S[SMi ]KL,1 [S]KL,1 / [SH]KL,1 / [SH2 ]KL,1

KL,1 {(kSH H) n1 / (kSH H) n1 [(KH SH2 H) n2 KL,1 {(kSH H) n1

[81]

[84]

(KH SH2 H) n2 / S(KH SHMi Mi ) li [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2

1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2

1 ( UT,SH / UT,SH2 / SUT,SHMi )

UT,SMj Å

li p2

/ S(KH SHMi Mi ) ] } rUL,T . / (kSH H) n1 [(KH SH2 H) n2

1 / [(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2 (KH SHMj Mj ) l j

1

/ S[SHMi ]KL,1 / S[SMi ]KL,1

Å

[(KH SH2 H) n2 / S(KH SHMi Mi ) li ] p2

1 ( UT,SH / UT,SH2 / SUT,SHMi )

/ S[SHMi ]KL,1 / S[SMi ]KL,1

[83]

(KH SH2 H) n2

[S]KL,1 / [SH]KL,1 / [SH2 ]KL,1

Å

[82]

The expressions for the total partial fractions U *T,SH , U *T,SH2 , U *T,SHMj , and U *T,SMj in Eqs. [76] – [79] can be derived in the same way as we have done before in deriving Eqs. [50], [40], [41], and [67], respectively. Combining these expressions with the ‘‘sum fractions’’ UT,T and ( UT,SH / UT,SH2 / SUT,SHMi ) given in Eq. [80] and [81] results in analytical expressions for the total binding of the individual species: UT,SH Å

Å

/ S(KH SHMi Mi ) li ] p2 rUT,T . / (KH SH H) n1 [(KH SH2 H) n2

/ S(KH SHMi Mi ) li ] p2 / S(KH SMi Mi ) £i

1 f (log KL,1 )d(log KL,1 )

Å

/ UL,SH2 / SUL,SHMi )

1 f (log KL,1 )d(log KL,1 )

UT,T Å UT,SH / UT,SH2 / SUT,SHMi / SUT,SMi Å

L,SH

(KH SMj Mj ) £ j n1

(KH SH H) / (KH SH H) n1 [(KH SH2 H) n2

rUT,T .

[85] [86]

/ S(KH SHMi Mi ) li ] p2 / S(KH SMi Mi ) £i

/ S(KH SHMi Mi ) li ] p2 / S(kSMi Mi ) £i }

The binding of individual species are thus obtained by taking fractions of fractions. Because KL,1 cancels from the numerator and denominator in the last equation above, this term can be placed in front of the integration. If we further multiply the numerator and ˜ SH ˜ L,1 , we may replace KSH and KSMi by K denominator by K and KH SMi , respectively. Integrating the results gives

AID

JCIS 4412

/

6g16$$$$68

09-10-96 12:17:07

ACKNOWLEDGMENTS D.G.K. gratefully acknowledges the Wageningen Agricultural University and The Royal Society for funding his stay in the Netherlands.

coida

AP: Colloid

50

VAN RIEMSDIJK ET AL.

REFERENCES 1. Stevenson, F. J., ‘‘Humus Chemistry: Genesis, Composition, Reactions.’’ Wiley, New York, 1994. 2. de Wit, J. C. M., van Riemsdijk, W. H., and Koopal, L. K., Environ. Sci. Technol. 27, 2015 (1993). 3. Bartschat, B. M., Cabaniss, S. E., and Morel, F. M. M., Environ. Sci. Technol. 26, 284 (1992). 4. de Wit, J. C. M., van Riemsdijk, W. H., and Koopal, L. K., Environ. Sci. Technol. 27, 2005 (1993). 5. Milne, C. J., Kinniburgh, D. G., de Wit, J. C. M., van Riemsdijk, W. H., and Koopal, L. K., Geochim. Cosmochim. Acta 59, 1101 (1995). 6. van Hal, R. E. G., Ph.D. dissertation, University of Twente, The Netherlands, 1994. 7. Kinniburgh, D. G., Milne, C. J., Benedetti, M. F., Pinheiro, J. P., Filius, J., Koopal, L. K., and van Riemsdijk, W. H., Environ. Sci. Technol. 30, 1687 (1996). 8. Dzombak, D. A., and Morel, F. M. M., ‘‘Surface Complexation Modeling: Hydrous Ferric Oxide.’’ Wiley, New York, 1990. 9. Milne, C. J., Kinniburgh, D. G., de Wit, J. C. M., van Riemsdijk, W. H., and Koopal, L. K., J. Colloid Interface Sci. 175, 448 (1995). 10. Koopal, L. K., van Riemsdijk, W. H., de Wit, J. C. M., and Benedetti, M. F., J. Colloid Interface Sci. 166, 51 (1994).

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11. Sips, R., J. Chem. Phys. 16, 490 (1948). 12. Benedetti, M. F., Milne, C. J., Kinniburgh, D. G., van Riemsdijk, W. H., and Koopal, L. K., Environ. Sci. Technol. 29, 446 (1995). 13. Gamble, D. S., Schnitzer, M., and Hoffman, I., Can. J. Chem. 48, 3197 (1970). 14. Bresnahan, W. T., Grant, C. L., and Weber, J. H., Anal. Chem. 50, 1675 (1978). 15. van Riemsdijk, W. H., Bolt, G. H., Koopal, L. K., and Blaakmeer, J., J. Colloid Interface Sci. 109, 219 (1986). 16. de Wit, J. C. M., van Riemsdijk, W. H., and Koopal, L. K., in ‘‘Humic Substances in the Global Environment and Implications on Human Health.’’ Elsevier, Amsterdam, 1994. 17. Rudzinski, W., Charmas, R., Partyka, S., Thomas, F., and Bottero, J. Y., Langmuir 8, 1154 (1992). 18. Rudzinski, W., Charmas, R., Partyka, S., and Bottero, J. Y., Langmuir 9, 2641 (1993). 19. Benedetti, M. F., van Riemsdijk, W. H., and Koopal, L. K., Environ. Sci. Technol. 30, 1805 (1996). 20. Tipping, E., Fitch, A., and Stevenson, F. J., Eur. J. Soil Sci. 46, 95 (1995). 21. Boyd, S. A., Sommers, L. E., Nelson, D. W., and West, D. X., Soil Sci. Soc. Am. J. 45, 745 (1981).

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AP: Colloid