INTERFACIAL APPLICATIONS IN ENVIRONMENTAL ENGINEERING - CHAPTER 11 pot

where SSA⫽ specific surface area [m2/g] and SC⫽ solids concentration [g/L],then approximations incorporating the monoprotic behavior assumptions for cal-culating pQ a1 and pQ a2values for

Effective Acidity-Constant Behavior Near Zero-Charge Conditions

Agency, Athens, Georgia, U.S.A

I INTRODUCTION

Current geochemical paradigms for modeling the solid/water partitioning ior of trace toxic ionic species at subsaturation mineral solubility porewater con-centrations rely on two fundamental mechanisms: (1) solid solution formationwith the major element solid phases present in the environment, and (2) adsorp-tion reactions on environmental surfaces Solid solution formation is the processleading to the substitution of a trace ion for a major ion in a natural solid phase(e.g., Ref 1) For example, solid solution formation between Cr3 ⫹and Fe(OH)3has been reported in the literature as a possible porewater solubility–limitingmechanism for dissolved Cr3 ⫹ This reaction can be described by

behav-nCr3 ⫹⫹ Fe(OH)3⇔ nFe3 ⫹⫹ Fe(1⫺n)Crn(OH)3

The second mechanism, the topic of this chapter, is generally believed to bemore widespread in environmental systems and is frequently described as theresult of surface complexation reactions between ionizable species (Mez ⫹) andreactive surface sites (⬎SOH) present on environmental solids, including ironoxides, manganese oxides, aluminum oxides, silicon oxides, aluminosilicates, andparticulate organic carbon For example, a reaction of the form

[⬎SOMe(z⫺1)⫹]⫽ concentration of complexed sites

Equation (1) differs from a solution counterpart in two ways: (1) Analogous tosurface protonation reactions, Eq (1) is a mixed concentration/chemical activityexpression Most practitioners make the assumption(s) that was (were) originallyapplied to surface protonation reactions that the activity coefficients for boundsites are equal and hence cancel out in the mass action quotient And (2), the

presence of the exponential Boltzmann expression (e ⫺∆G(excess)/RT) The Boltzmannexpression as commonly used is generally predicated on the assumption that anyexcess energy is primarily electrostatic in nature (i.e.,∆Gexcess ⫽ ∆Gelectrostatic) and

that this energy results from moving mobile ions between bulk solution (where

∆Gelectrostatic ⫽ 0) and the interfacial region (where ∆Gelectrostatic≠ 0) (e.g., see Ref 5)

By inspection of Eq (1), one can observe that there is an inherent competitionfor reactive bound sites between metal ions and the hydrated proton Pragmati-cally speaking, an inspection of Eq (1) leads to a predicted “release” of bound(i.e., surface-complexed) metal ions when a solid/liquid system is acidified Due

to recognition of the inherent competition for bound sites by the hydrated protonand fundamental uncertainties in our ability to describe surface acidity reactions,two publications [6,7] concluded that the majority of uncertainty in our ability

to model ionic contaminant adsorption behavior was due to limitations in ourunderstanding of surface acidity behavior Hence, a fundamental understanding

of the protonation behavior of reactive sites on environmental surfaces is a uisite to a better understanding of the partitioning behavior of the ionizable spe-cies of toxicological interest

prereq-Most researchers use the two-pK surface complexation model for describingthe protonation behavior of environmental hydrous oxide adsorbents They gener-ally assume that bound surface sites can exist in one of three protonation condi-tions:⬎SOH2⫹,⬎SOH, and ⬎SO⫺ Mass action expressions commonly used forquantifying the equilibration among protonated surface sites in response to thechemical activity of the hydrated proton are:

For both computational convenience and as a result of experimental difficulties

in measuring∆Gelectrostatic, a number of authors adapted procedures previously

ap-plied to polyelectrolytes/latex particles [12–18] and rearranged Eqs (2) and (3)into forms that are more amenable to computation from experimental data:

methodol-cance to the present study is that variations of Q a1 and Q a2as functions of chargedensity, pH, and ionic strength can lend insight into the nature of those energiescontributing to∆Gexcess.

Equations (1) to (5) are generally utilized with the assumption that the excesselectrostatic Gibbs free energies for these systems (∆Gexcess) are reasonably ap- proximated by integer multiples of F Ψ (where F equals Faraday’s constant and

Ψ is the electrostatic potential in the interfacial region) As will be demonstrated

in the next section, there are theoretical reasons to question this assumption

A Origin of the Charging-Energy Term

Chan et al [9] defined the electrochemical potentials (u) of the surface reacting

species in Eqs (2) and (3) in the following way:

where γ⬎SOHxis the activity coefficient for surface site ⬎SOHx, e is the charge

of the electron, and k is the Boltzmann constant The electrostatic component of the electrochemical potential of the interfacial hydrated proton (eΨ) in Eq (6a)has been discussed extensively in the literature and results from moving mobile

(whereΨ ≠ 0; e.g., see Ref 5) The electrostatic components of the cal potentials of the ionized surface sites in Eqs (6c) and (6d) can be viewed asbeing representative of the charging energies associated with creating a net charge

∑(u oproducts)⫺ ∑(u oreactants), K⫽ e ⫺∆Go/RT, and one assumes that the bound site activitycoefficients in Eqs (6b) to (6d) equal one another, then the electrostatic compo-nent of∆G in the Boltzmann expression in Eqs (4) and (5) (∆Gelectrostatic) as derived from these electrochemical potentials should be 2eΨ (on a per-ion basis) or 2FΨ (on a molar basis) rather than the traditional value of eΨ or FΨ Specifically,

with this thermodynamic analysis of surface protonation/deprotonation reactionsoccurring in the absence of surface charge neutralization by counterelectrolyte

being attributable to moving a mobile ion between neutral bulk solution and the

charged interfacial region and one FΨ resulting from the creation of a site with

a unit charge of “⫾e” under conditions of constant potential Ψ.

The present author [19] further examined charging energies by integrating a

the particle charge,ε ⫽ the aqueous dielectric constant, ε0⫽ the permittivity of

free space, and r ⫽ the particle radius) from Q to Q ⫾ e Specifically, Ref 19

∆Gcharging⫽ (Q ⫾ e)2⫺ Q2

8πεε0r

(assuming an integration constant of zero) It was also demonstrated that when

Q ⬎⬎ e, then ∆Gcharging ⬇ ⫾eΨ This analysis was predicated on the assumption

that the surface region where charged sites are located is impenetrable to electrolyte ions Based on this analysis,∆Gelectrostaticin Eqs (2) to (5) also should

counter-equal 2e Ψ (on a per-ion basis) or 2FΨ (on a molar basis) under constant-potential

conditions

The present author [19] also examined circumstances where electrolyte ionscan penetrate the surface region and partially neutralize the charge associated

with the created charged site Given that the fraction of net surface charge ized by electrolyte ions is assigned a value ofτ (where τ ranges from zero to 1

integral of

∆Gcharging⫽ (Q ⫾ [1 ⫺ τ]e)2⫺ Q2

8πεε0r

When Q ⬎⬎ e, the charging energy was found to be approximated by ∆Gcharging⬇

⫾(1 ⫺ τ)eΨ If one then derived a mass action expression from the chemical

potentials of the reacting species, the total electrostatic expression in the mann term (∆Gelectrostatic) of the respective mass action expressions given in Eqs.(2) to (5) was estimated to be (2⫺ τ)FΨ (on a molar basis) or (2 ⫺ τ)eΨ (on

Boltz-a per-ion bBoltz-asis) FinBoltz-ally, through extensive computer simulBoltz-ations, it wBoltz-as Boltz-alsoobserved that (2⫺ τ) approaches a value of 1 at high charge densities for allionic strengths (thereby supporting the historical mass action formulations) How-ever, it also was predicted that (2⫺ τ) would significantly deviate from a value

of 1 at low-charge conditions In essence, it was predicted that charging energies

will lead to increased values of calculated pQ a1 and pQ a2terms in the pHzpcregionthat is inconsistent with conventional diffuse layer modeling

B Significance of Aggregation-Derived

Neutral Size Sequestration

Traditional approaches for using the pHzpc extrapolation procedure in biproticsystems have relied on the assumption of monoprotic behavior both above andbelow the pHzpc Specifically, below the pHzpc the concentration of negativelycharged sites is assumed to be insignificant, and above the pHzpcthe concentration

of positively charged sites is assumed to be insignificant The rigorous definitions

for pQ a1 and pQ a2are given by

(where SSA⫽ specific surface area [m2/g] and SC⫽ solids concentration [g/L]),then approximations incorporating the monoprotic behavior assumptions for cal-

culating pQ a1 and pQ a2values for titrimetric data are given by

pQ a1⫽ pH ⫺ log(σtot⫺ σ)

and

As will be demonstrated in Section III, the assumptions of monoprotic ior below and above the pHzpcwith the pHzpcextrapolation procedure leads to an

behav-underestimate of the true pQ a1 values and an overestimate of the true pQ a2values

in the pHzpcregion As a first approximation, these errors are the result of ing that [⬎SOH] is directly proportional to (σtot ⫺ σ) (below the pHzpc) and

assum-⫺(⫺σtot⫺ σ) (above the pHzpc) and that [⬎SOH2⫹] is directly proportional toσ(below the pHzpc) and that [⬎SO⫺] is proportional to σ (above the pHzpc) Insummary, these approximations suffer from an error that increases with proximity

to the pHzpcand is the result of simultaneously overestimating [⬎SOH] and estimating charged site concentrations in the pHzpcregion

under-It is hypothesized here that there exists an experimental artifact that can have

a similar effect Specifically, it is not uncommon for an experimenter to observesubstantial aggregation in titrations at pH conditions adjacent to the pHzpc Thisphenomenon may be responsible for the widely reported observed hysteresis inforward and backward titrations of hydrous oxide slurries Secondly, it is notunreasonable to believe that aggregation will render some sites inaccessible to

a given titrant (at least within the equilibration times commonly used in theseexperiments) Finally, given the local acid–base disequilibrium conditions thatexist prior to complete mixing of a titrant addition to a slurry in an experimentalvessel, it is hypothesized here that neutral and oppositely charged sites will tend

to be preferentially “buried” during the aggregation process Qualitatively, and

in contrast to charging energy phenomena, aggregation-derived sequestration oftitrable sites in the pHzpcregion is predicted to cause the same type of error ob-served with the pHzpcextrapolation procedure That is, this error is hypothesized

to simultaneously decrease pQ a1 estimates and increase pQ a2 estimates in the

pHzpcregion

The remainder of this chapter will focus on: (1) developing a method to ate simulated titrimetric data of known accuracy (using 17-digit double-precisionGW-BASIC [21]), (2) developing two alternative methods to the pHzpcextrapola-

gener-tion procedure for extracting Q a values from titrimetric data, (3) assessing allthree methods with simulated data, and, finally, (4) applying these methods to

titrimetric data published in the literature for the purpose of identifying possiblecharging energy and/or aggregation-derived titrable site sequestration contribu-tions to effective acidity-constant behavior.

II METHODS

A A Method for Simulating Titrimetric Data

If one combines Eqs (4), (5), (7), and (8), the following expression for a biproticsystem can be derived:

Expression (9) is particularly useful; among other things, it may be used to late titrimetric data For a given system with specified values for temperature,ionic strength, σtot, Ka1 , and K a2 and assuming traditional diffuse layer modelbehavior, one can ultimately estimate the hydrogen ion activities required to yield

simu-a given vsimu-alue ofσ with the quadratic solution For example, for a given value

ofσ, one can first calculate a value of Ψ using the Gouy–Chapman 1-dimensionalsolution to the Poisson–Boltzmann equation; e.g., at 25°C,

Ψ ⫽ sinh⫺1σ/{0.1174 * I1/2}

19.46 * z Values for Q a1 and Q a2can then be generated by

Q a1 ⫽ K a1 e F Ψ/RT and Q a2 ⫽ K a2 e F Ψ/RT

Finally, with the substitutions a⫽ (σtot⫺ σ), b ⫽ ⫺Q a1 σ, and c ⫽ ⫺Q a1 Q a2(σtot⫹σ), the hydrogen ion activity required to achieve a given value of σ can be calcu-lated by

σtot⫽ ⫺{[⬎SOH] ⫹ [⬎SO⫺]}F

{SSA * SC}

Hence, Q avalues can be extracted from titrimetric data for a monoprotic system

with the expression Q a ⫽ σaH⫹/(σtot⫺ σ) In contrast to biprotic systems, these

values for Q acan be obtained directly from titrimetric data without the mation errors in relating [⬎SO⫺] and [⬎SOH] to σtotandσ

approxi-Equation (9) also may be used to extract Q a1 and Q a2values from experimentaldata derived from a biprotic system By inspection of Eq (9), the reader can

discern that for any given data point characterized by aH(⫹)andσ (and where σtot

is known), one cannot solve explicitly for Q a1 and Q a2because there exists onlyone equation [Eq (9)] and two unknowns In theory however, Eq (9) can besolved for two unknowns by using two adjacent data points in a titration curve

if the effective acidity constants can be assumed to remain nearly constant forthese two points Although only two data points are required with this procedure,

the present author [20] found that solving for values of Q a1 and Q a2twice usingthree consecutive data points and averaging the values tended to minimize ex-

treme estimates of Q abehavior The procedure of solving Eq (9) twice with threeconsecutive data points and averaging the results will be used in this work and

will be termed the direct substitution procedure.

One may also take partial derivatives of Eq (9) with respect to aH(⫹) andσand obtain the following relationship:

Average effective pQ values can then be calculated by averaging the values

ob-tained from twice solving two equations for two unknowns

In summary, this work will involve using Eq (9) to generate simulated titrationcurves at various ionic strengths for a biprotic surface (with intrinsic acidity con-stants of 10⫺6and 10⫺8) using the Gouy–Chapman charge/potential relationship

an accuracy of 17 digits [21] Data obtained from the simulated curves will then

be subjected to the conventional pHzpcextrapolation procedure and the tion and differential methodologies described earlier for the purpose of assessing

substitu-the accuracy of substitu-these methods for extracting Q a1 and Q a2values from the lated experimental data Lastly, these extraction methodologies will be applied

simu-to experimental data obtained from the peer-reviewed literature for the purpose

of interpreting anomalous pQ behavior in the pH zpcregion within the context ofpossible charging-energy and aggregation-derived site sequestration phenomena

III RESULTS

A Results from Simulated Data

Figure 1 illustrates simulated pQ a1values as a function of ionic strength derived

for a Gouy–Chapman surface with intrinsic acidity constants of K a1⫽ 1E-6 and

K a2⫽ 1E-8 The maximum site density for this surface was set at 0.32 C/m2,and the temperature was held at 298 K with these simulations The “fictional”

104M ionic strength simulations were used to saturate the Gouy–Chapman trostatic term (i.e., the maximum estimated surface potential at an “ionic strength”

elec-of 1E4 M was estimated to be⫾0.0004 V) The reader should note that the pQ a1

values for ionic strengths 1E-1, 1E-2, and 1E-3 M display logistic or S-shapedcurves as functions of charge density; these shapes are more characteristic of a

diffuse layer model of the interface The pQ a1 values at an ionic strength of1E-1 M generate a more “linear” curve and, hence, illustrate a possible situation

Gouy–Chapman diffuse layer surface in aqueous solution Maximum charge density⫽0.32 C/m2, T ⫽ 298 K, pK a1 ⫽ 6, and pK a 2⫽ 8

for justifiably using a constant-capacitance-charge/potential relationship [22]

Al-though not shown here, the pQ a2values displayed identical curves that were offset

from the pQ a1values by 2 pK units

Figure 2 displays simulated titration data for the biprotic Gouy–Chapman face described inFigure 1.These data were generated by inserting the previously

sur-estimated pQ a1 and pQ a2 values used to construct Figure 1 into the quadraticequation [Eq (9)] and solving for the required hydrogen ion activity

Figure 3depicts estimated values of pQ a1 and pQ a2extracted from the lated data at an ionic strength of 1E4 M displayed in Figure 2 It is gratifying

simu-to note that the substitution and differential procedures yielded effective acidity

constants comparable to the “true” values for pQ a1below the pHzpc and for pQ a2

above the pHzpc The pHzpcextrapolation methodology suffered from significanterror in the pHzpcregion due to the assumption that the concentrations of oppo-sitely charged sites both above and below the pHzpcwere insignificant Estimated

1 using the quadratic equation [Equation (9)]; the “fictional” ionic strength of 1E-4 molarwas performed to minimize electrostatic effects

FIG 3 Comparison of “true” pQ values with the pH zpcextrapolation procedure, tion, and differential methodologies for estimating effective acidity constants from the1E4 M simulated data presented inFigure 2.Note the errors in the pHzpcregion usingthe pHazpcextrapolation procedure The substitution and differential methodologies yield

substitu-significant deviations for pQ a 2below the pHzpc(possibly due to round-off errors)

values for pQ a2below the pHzpcwith the substitution and differential proceduresdisplayed diminished accuracy, possibly due to round-off errors in the algorithmused to estimate these numbers

Figure 4compares the results from the three extraction methodologies as plied to the 1E-3 M simulated data displayed inFigure 2.In contrast to the resultsdisplayed in Figure 3, the pHzpcmethodology yields a slightly superior accuracy(when compared to its relative performance in Figure 3) The marginally im-proved performance of the pHzpcextrapolation procedure at an ionic strength of1E-3 M (when compared with the substitution and differential methodologies)can be ascribed to the fact that the performance of the two other methodologies

points

The results displayed in Figures 3 and 4 tend to support a contention that there

is no perfect methodology for extracting pQ a values from titrimetric data for