10092
J. Phys. Chem. B 2008, 112, 10092–10100
Competitive Cd2+/H+ Complexation to Polyacrylic Acid Described by the Stepwise and Intrinsic Stability Constants Calin David,† Encarnacio´ Companys,† Josep Galceran,† Josep Lluı´s Garce´s,† Francesc Mas,‡ Carlos Rey-Castro,† Jose´ Salvador,† and Jaume Puy*,† Departament de Quı´mica, UniVersitat de Lleida (UdL), AVenida RoVira Roure, 191. E-25198 Lleida, Spain, and Departament de Quı´mica Fı´sica, Facultat de Quı´mica, UniVersitat de Barcelona (UB) and Institut de Quı´mica Teo`rica i Computacional (IQTCUB), C/ Martı´ i Franque`s, 1. E-08028 Barcelona, Spain ReceiVed: March 25, 2008; ReVised Manuscript ReceiVed: June 2, 2008
Stepwise constants can be used to describe competitive proton and metal binding to macromolecules with a large number of sites. With the aim of accessing information on the microscopic binding model, we report an expression that connects the stepwise constants to the site-specific metal constants. This expression holds for a very general complexation model including heterogeneity, interactions, and chelate complexation. Assuming bidentate binding of the Cd ions to adjacent carboxylate groups in poly(acrylic acid), stepwise and intrinsic stability constants for proton and cadmium binding were estimated from the experimental data. Intrinsic values were split into specific and electrostatic contributions (by means of the Poisson-Boltzmann equation under cylindrical geometry). Free of the electrostatic contribution, the remaining Cd binding energy showed almost no dependence on the coverage and ionic strength, and the corresponding average values allowed for a reasonable reproduction of raw binding data. Small systematic discrepancies from the homogeneous behavior are critically discussed. 1. Introduction Binding equilibria of ions to heterogeneous macromolecules play a crucial role in the circulation of metals in natural systems. Many interpretative tools have been used to describe the data on ion binding to natural heterogeneous particles such as conditional stability constants, adsorption isotherms, and affinity spectra.1–3 In all of these cases, the binding energy can be split into electrostatic and specific contributions.4 However, even in the relatively simple case of chemically homogeneous polyions, the application of electrostatic models and the description of competitive binding for different metal-to-ligand stoichiometric ratios are not trivial.5–7 Although some effort has been devoted to analysis of the agreement between the results provided by these different tools,8,9 comparison of the respective results is still sometimes difficult because of the lack of a theoretical background: different adsorption isotherms contain parameters with different units and physical meanings, different methods of handling of electrostatic binding (Poisson-Boltzmann with different geometries, Donnan equilibrium with different volume expressions, etc.) are used, and experiments are conducted in different concentration ranges. Altogether, these circumstances lead to different results, especially when the ligands are heterogeneous, i.e., contain complex mixtures of chemical species with different functional groups to which the ions bind. The description of the binding data is still more difficult for competitive systems, the typical situation in natural media. Actually, metal binding sites usually exhibit acid-base behavior, and metal binding data are strongly dependent on pH, not only because of the change of the electrostatic contribution by changing the pH, but also because of the increasing amount of * Corresponding author. E-mail:
[email protected] † Universitat de Lleida (UdL). ‡ Universitat de Barcelona (UB) and Institut de Quı´mica Teo ` rica i Computacional (IQTCUB).
metal binding energy that has to be expended in extracting the proton as the pH decreases. In addition to all of these phenomena, there is another effect that is usually not explicitly taken into account in the description of the competitive binding: the different stoichiometry of the metal with respect to that of the proton or the different stoichiometries of different metals. Metal sites can consist of more than one functional group to which protons bind. Under such conditions, proton sites are not metal binding sites, the maximum coverage is ion-dependent, and the physical meanings of parameters within theoretical frameworks that do not consider this effect explicitly are lost, rendering the parameters “apparent” or “effective”. Stepwise constants correspond to equilibrium constants of increasing sequential occupation of macromolecular sites, regardless of the specific sites to which the complexing ion is bound.10 Recently, we reported11 a simple expression allowing for the computation of stepwise stability constants for competitive systems in which the ligand has a large number of sites. The use of these constants in the description of the binding data is not restricted to a fixed stoichiometry, thus providing a powerful technique. Furthermore, information on the microscopic binding model can be gained by obtaining the intrinsic stability constants (where the configurational factor has been removed) and by using appropriate statistical mechanics techniques.12–14 This work provides a general relationship between the stepwise and site-specific constants so that the stepwise values can be used to access information on the microscopic binding model. Although extensive work on the binding properties of complexing ions has been done in different fields ranging from biochemistry to heterogeneous catalysis, to the best of our knowledge, a similar relationship with such generality has not been reported. Section 3 describes the experimental methods and conditions, as well as the approach by which stepwise constants are used to characterize the Cd2+/H+ binding to
10.1021/jp802571f CCC: $40.75 2008 American Chemical Society Published on Web 07/29/2008
Competitive Cd2+/H+ Complexation to PAA
J. Phys. Chem. B, Vol. 112, No. 33, 2008 10093
poly(acrylic acid) (PAA). In section 4, detailed analysis of the intrinsic constants and correction for the electrostatic contribution to the binding energy, by means of the Poisson-Boltzmann equation, allow us to draw some conclusions about the electrostatic and specific contributions to the binding of these ions to PAA. 2. Methodology: Determination of the Stepwise Stability Constants for a Macromolecular Ligand with a Large Number of Sites 2.1. Theoretical Background: Stepwise Stability Constants. Consider a macromolecule (P) with sH sites for the protons and sM sites for metal ions. These numbers, sH and sM, do not need to be equal, so that different stoichiometric relationships (chelate binding) can be included in our formalism. The sequential complexation of protons and metal ions to the macromolecule can be schematized as Ki,j
HiMj-1P + M y\z HiMjP
(1)
where HiMjP denotes the stoichiometric species with i bound protons and j bound metal ions, regardless of the particular sites occupied by these ions. The corresponding set of equilibrium constants, the so-called stepwise constants, can be labeled using the two subscripts, i and j, so that
Ki,j )
cHiMjP cHiMj-1PcM
(2)
With the pertinent changes, a similar definition can be used for the stepwise constants for proton binding, Ki,Hj, indicating the equilibrium constant for the binding of the ith proton to species with i - 1 protons and j metal ions. For the sake of simplicity, we omit the superscript M when the constants refer to metal binding, and we only quote superscript H explicitly to refer to proton binding. From the fundamental equilibrium relationships, eq 2, the average number of bound metal ions, νM (a quantity that is experimentally available if the total concentration of macromolecule is known), can be expressed as a function of the free metal concentration as sH sM
νM )
∑∑
jβi,jcHicMj
i)0 j)0 sH sH
(3)
∑ ∑ βi,jcHicMj i)0 j)0
where βi,j ) Ki,1Ki,2 · · · Ki,j is the so-called overall stability constant, corresponding to the stability constant of the process of binding i protons and j metal ions to the naked macromolecule. The denominator in eq 3 is the so-called binding polynomial or macrocanonical partition function sH sM
Ξ)
∑ ∑ βi,jcHicMj
(4)
i)0 j)0
so that eq 3 can also be rewritten as
νM )
(
∂ ln Ξ ∂ ln cM
)
the stepwise constants in terms of some microscopic parameters (binding and interaction free energies, conformational properties, etc.).13,17 However, in many macromolecules of interest, the number of complexing sites, and thus the number of stepwise constants, is very large, and the nonlinear fitting of eq 3 becomes highly dependent on the experimental error.15 Recently, a very simple method for determining stepwise stability constants for ligands with a high number of sites was suggested.18 Applying the maximum term theorem of statistical mechanics,18,19 Ki,j can be expressed as
lim log Ki,j ) [-log cM]νH)i,νM)j
sMf∞ sHf∞
where [cM]νH)i,νM)j indicates the free metal ion concentration when the mean occupation numbers of protons and metal ions are νH ) i and νM ) j, respectively. Notice that the right-hand side of eq 6 depends only on information that can be directly obtained from the binding curve, so that eq 6 adds a new physical meaning to the binding curve: In a plot of pM ) -log cM as a function of νM and νH, the value of the ordinate at the point νM ) j and νH ) i provides log Ki,j directly for a ligand with a large number of sites. Analogously, log Ki,Hj can be obtained by replacing pM with pH. Although sM f ∞, sH f ∞ can be seen as a very restrictive condition, it has been shown that, for several complexation models18 (heterogeneous systems and systems with interactions between occupied sites), the application of eq 6 is quite accurate for systems with sX g 20. Certainly, this is a very low number of sites for many polyelectrolytes of interest where characteristic numbers of sites are on the order of 102-104. Not all of the stepwise proton and metal binding constants are independent. They are related by closure relationships because of the irrelevance (in the change in Gibbs energy) of the order in which i′ protons and j′ metal ions bind to species HiMjP to generate species Hi+i′Mj+j′P i′
∑
m)1
j′
H log Ki+m,j +
∑
j′
log Ki+i’,j+n ) ... )
n)1
The standard procedure for the determination of βi,j consists in fitting eq 3 to the experimental binding data.10,15,16 In this task, techniques of statistical mechanics can help in expressing
∑ log Ki,j+n +
n)1
i′
H ∑ log Ki+m,j+j′
(7)
m)1
Only two limiting situations have been made explicit in this expression: (i) all of the necessary protons are bound before the binding of the necessary additional metal ions begins and (ii) all of the metal ions are initially bound. Obviously, other combinations are possible, leading to additional closure conditions. 2.2. Relationship between Stepwise and Site-Specific Stability Constants. Even for the simplest case of homogeneous and independent sites, the stepwise constants, Ki,j, depend on the stoichiometric step. This dependence stems from the fact that, apart from the Gibbs energy of the binding to a specific site, an additional entropic configurational term (which considers the number of chemical species with i protons and j (or j - 1) metal ions differing in the particular occupied sites) also participates in the stepwise constants. One can split the stepwise constants into a configurational term and a remaining term that is called the intrinsic binding constant. Following previous definitions,11 the intrinsic stability constants can be defined as
(5) cH
(6)
Ki,j )
gi,j K gi,j-1 i,j,int
(8)
where gi,j indicates the number of ways of distributing i protons and j metal ions in the naked macromolecule. Clearly, gi,j will
10094 J. Phys. Chem. B, Vol. 112, No. 33, 2008
David et al. greater than the set of HiMjP subspecies, given that a particular HiMjP subspecies with a fixed position for metal ions and protons can be generated from j different HiMj-1P subspecies: those obtained by just extracting each one of the j metal ions present in the given HiMjP subspecies. (In Figure 1, R2 can be generated both from τ6 and from τ7.) Thus, the total number of different metal free sites in the overall set of HiMj-1P subspecies is j × gi,j. We refer to these free metal sites as potential sites, because the total number of metal ions that can be bound to these free sites is usually lower than the total number of free sites (i.e., in some cases, metal binding to some of these potential sites can prevent simultaneous binding to other potential sites incompatible with them). We arbitrarily index these potential sites from p ) 1 to p ) jgi,j. We use the variable kp to represent the microscopic stability constant for the binding of M to the potential site p. To keep track of the subspecies connected by a given kp, we introduce the function f(p) for the index of the “original” HiMj-1P subspecies and the function h(p) for the index of the “arriving” HiMjP subspecies [e.g., in Figure 1, f(3) ) 6 and h(3) ) 2]. Thus
cRh(p) ) kpcτf(p)cM
(10)
Summing over all potential sites gives jgi,j
∑
p)1
jgi,j
∑ kpcτ
cRh(p) )
p)1
cM
(11)
f(p)
Each subspecies Ra is included j times, so from eq 9, we obtain jgi,j
Figure 1. Scheme of the subspecies of HMP (denoted as τ,..., τ12) and HM2P (denoted as R1, R2, R3) for a macromolecular chain of length n ) 5. Arrows indicate the occupation of the potential sites in HMP species, and kp is the corresponding site-specific constant.
be dependent on the stoichiometries of the proton and metal binding. The practical use of eq 8 requires the assumption (or knowledge) of a given stoichiometry for metal binding, but even then, the relationship between the stepwise and site-specific constants has to be clarified. The concentration of HiMjP can be expressed as the sum of the concentrations of different species or instances (which we henceforth label as subspecies) differing in the position of the i protons and j metal ions. For the sake of notation simplicity, we refer to the subspecies of HiMj-1P as τ1, τ2,..., τgi,j-1 and to the subspecies of HiMjP as R1, R2,..., Rgi,j, with the indices following any arbitrary order of the subspecies. For instance, Figure 1 shows an example where sH ) 5 and sM ) 2 (i.e., sH * sM)with g1,1)12 subspecies of species HMP and only g1,2 ) 3 subspecies for HM2P. If cRa represents the concentration of a given subspecies Ra, then gi,j
cHiMjP )
∑ cR
a)1
a
∑ cR
p)1
h(p)
) jcHiMjP
(12)
Thus, by combining the previous expressions, we obtain gi,j
Ki,j )
cHiMjP cMcHiMj-1P
)
∑
a)1
jgi,j
cRa
cMcHiMj-1P
)
∑ kpcτ
p)1
cM
f(p)
jcMcHiMj-1P
)
jgi,j cτf(p) 1 kp j p)1 cHiMj-1P
∑
(13)
Expression 13 gives the stoichiometric constants in terms of the microscopic ones (i.e., those corresponding to the potential sites), and it indicates that Ki,j is the inverse of j times the sum of all of the microscopic stability constants of all of the potential sites of the set of species HiMj-1P, each microscopic stability constant being weighted by a factor representing the fractional concentration of the subspecies to which this potential site belongs relative to the total concentration of HiMj-1P species. Equation 13 holds for sites with different affinities (heterogeneous complexation), chelate binding, and interactions between bound species. In terms of the intrinsic constants, eq 13 can be rewritten as
(9)
Notice that different subspecies can have different numbers of free metal sites (τ1 has one potential site for the addition of the second M in the example of Figure 1, whereas τ2 has none). Through the conceptual addition of a metal ion to any of the free sites of a given subspecies of HiMj-1P, we generate one of the possible subspecies in the species of HiMjP. For instance, in Figure 1, from τ6 we generate R2. However, the total number of different free metal sites in the set of HiMj-1P subspecies is
cτf(p) gi,j-1 jgi,j kp Ki,j,int ) jgi,j p)1 cHiMj-1P
∑
(14)
where eq 8 has been used. For the particular case of homogeneous complexation, the concentration of all the subspecies with a common number of protons and metal ions bound is the same, so cτf(p)/cHiMj-1P ) 1/gi,j-1. Additionally, all of the microscopic sites have the same affinity k, so
Competitive Cd2+/H+ Complexation to PAA
J. Phys. Chem. B, Vol. 112, No. 33, 2008 10095
Ki,j,int ) k
(15)
Equation 15 indicates that, for homogeneous complexation, regardless the stoichiometry of the metal binding, the intrinsic constants are independent of the complexation step (i.e., i and j) and are equal to the binding energy of the metal to a specific site. Let us particularize the preceding discussion to a competitive system consisting of a macromolecular ligand and two complexing ions, proton and metal, and let us consider that the metal ions bind to m adjacent functional groups of a linear macromolecule (with a total of n groups). The number of species differing in the specific sites occupied by the i bound protons or the j bound metal ions is
gi,j )
(n - mj + j)! i ! j ! (n - mj - i)!
(16)
The entropic factor in eq 16 arises from the number of combinations without repetition of the n - mj - i free monomers, the i bound protons (the stoichiometry of protons is taken as 1:1), and the j bound metal ions. Equation 16 predicts 12 subspecies for HMP and 3 for HM2P (Figure 1) for a particular chain of length n ) 5 and m ) 2. According to eq 13, K1,2 can be written as
K1,2 )
[
cτ1 cτ3 cτ6 1 k1 + k2 + k3 + 2 cHMP cHMP cHMP k4
cτ7 cHMP
+ k5
cτ10 cHMP
+ k6
cτ12 cHMP
]
(17)
where kp is the site-specific constant of each potential site. Notice that, although the factors cτf(p)/cHiMj-1P are dependent on the affinities of the occupied potential sites, eq 17 holds for any values of the set of site-specific constants, thus indicating the generality of this equation. In particular, we do not assume that k1 ) k2, so that interactions between occupied sites are allowed. For the particular case of homogeneous complexation (in which all of the potential sites have the same affinity kM), the weighting factor of each microscopic constant is just 1/12 (there are 12 subspecies of HMP)
K1,2 ) kM
1 4
(18)
and according to eqs 14 and 15
K1,2,int ) K1,2
g1,1 1 12 ) kM ) kM g1,2 4 3
(19)
That is, the intrinsic constant equals the site-specific one. 3. Experimental Section 3.1. Materials. Polyacrylic acid (PAA), with an average molecular weight of 250 kD, was purchased from Aldrich and used without further purification. KNO3 (Fluka, TraceSelect) was used as the supporting electrolyte. The titrating solutions were 0.1 M standard aqueous solutions of HNO3 and KOH (Merck Titrisol) and 0.0333 M Cd(NO3)2 solutions prepared from the solid product (Merck, analytical grade). Ultrapure water (Milli-Q Plus 185 System from Millipore) was employed in all experiments. Purified water-saturated nitrogen N2(50) was used for deaeration. 3.2. Methods. The potentiometric measurements were carried out coupling an Orion pH/ISE meter with a Metrohm Dosimat dispenser. The entire titrating setup was controlled by a homemade program running on a personal computer. The
samples were placed in a double-walled potentiometric glass cell thermostated at 25 °C. N2 bubbling and soda lime traps were used throughout to prevent CO2 contamination. The potential between the ion-selective electrode (pH or Cd electrode) and a Ag/AgCl reference electrode was measured and recorded after a drift criterion of
