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Adsorption of organic pollutants

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Adsorption of organic pollutants

GEOCHEMICAL MODELS

    FOR SEDIMENT-SEAWATER INTERACTIONS

     Solute

     Solvent Surface

    Peer Børretzen & Brit Salbu

    Laboratory for Analytical Chemistry Institute for Chemistry and Biotechnology

    @ Agricultural University of Norway

    1432 Ås, Norway

    (peer.borretzen@ikb.nlh.no)

    30. April 2012

TABLE OF CONTENTS

    1 OBJECTIVES ......................................................................................................................................... 1

    2 INTRODUCTION ................................................................................................................................... 1

    2.1 SPECIATION OF MICROPOLLUTANTS ..................................................................................................... 1 2.2 SEDIMENT-SEAWATER INTERACTIONS ................................................................................................. 2 2.3 KINETIC INFORMATION ....................................................................................................................... 5 3 SORPTION OF CHARGED TRACE METALS AND RADIONUCLIDE SPECIES ........................... 5

    3.1 EMPIRICAL SORPTION MODELS ............................................................................................................ 6

    3.1.1.1 Distribution coefficients ........................................................................................................................ 6

    3.1.1.2 Langmuir isotherm ................................................................................................................................ 6

    3.1.1.3 Freundlich isotherm............................................................................................................................... 6

    3.1.1.4 General partitioning equation ................................................................................................................. 7

    3.2 SURFACE COMPLEXATION MODELS ...................................................................................................... 7

    3.2.1 Interface (electrostatic sorption) models......................................................... 7

    3.2.1.1 Constant Capacitance Model (CCM) ...................................................................................................... 7

    3.2.1.2 Diffuse Layer Model (DLM) ................................................................................................................. 8

    3.2.1.3 Basic Stern Model (BSM)...................................................................................................................... 8

     3.2.1.4 Triple Layer Model (TLM) .................................................................................................................... 8

    3.2.1.5 Nernstian Stern Model (NSM) ............................................................................................................. 10

    3.2.1.6 Three Plane Model (TPM) ................................................................................................................... 10

    3.2.1.7 Charge Distribution Model (CD).......................................................................................................... 10

    3.2.1.8 Non-electrostatic surface complexation model (NESCM) ..................................................................... 10

    3.2.2 Complexation (chemisorption) models .......................................................... 10

    3.2.2.1 Two state model (1 pK) ....................................................................................................................... 11

    3.2.2.2 Three state model (2 pK) ..................................................................................................................... 11

    3.2.3 Numerical solution of the equilibrium problem ............................................... 11 3.3 ION EXCHANGE ................................................................................................................................ 13

    3.4 SORPTION OF TRACE METALS ON NATURAL ORGANIC MATTER ............................................................ 15

    3.4.1.1 Mixture Model (MM; Mattigod & Sposito 1979) .................................................................................. 15

    3.4.1.2 Gaussian Model (GM; Perdue & Lytle 1983) ....................................................................................... 15

    3.4.1.3 Model V (Tipping & Hurley 1992) ...................................................................................................... 15

    3.5 SPECIATION COMPUTER CODES.......................................................................................................... 16

    3.5.1.1 PHREEQC (USGS; Parkhurst 1995) .................................................................................................... 16

    3.5.1.2 CHESS (Ècole des Mines de Paris; van der Lee & de Windt 1999) ....................................................... 16

    3.5.1.3 HYDRAQL (Stanford University; Papelis et al. 1988) .......................................................................... 16

    3.5.1.4 EQ3/6 (Lawrence Livermore National Laboratory; Wolery 1992) ......................................................... 16

    3.5.1.5 MINTEQA2 (USEPA; Allison et al. 1991) ........................................................................................... 17

    3.5.1.6 FITEQL (Herbelin & Westall 1996) ..................................................................................................... 17

    4 ADSORPTION OF ORGANIC SOLUTES .......................................................................................... 17 4.1 HYDROPHOBIC ORGANIC COMPOUNDS ............................................................................................... 17 4.2 HYDROPHOBIC IONISABLE ORGANIC COMPOUNDS .............................................................................. 18 4.3 HYDROPHOBIC IONIC ORGANIC COMPOUNDS ...................................................................................... 18 5 KINETIC MODELS ............................................................................................................................. 19

    6 DISCUSSION ........................................................................................................................................ 20

    6.1.1.1 The importance of organic matter when complexing trace-metals in estuarine-marine systems ............... 22

    6.1.1.2 Empirical vs. SCM model .................................................................................................................... 23

    6.1.1.3 Guidelines given by Dzombak et al. (1987) .......................................................................................... 23

    6.1.1.4 Guidelines given by Davis & Kent (1990) ............................................................................................ 24

    6.1.1.5 Which SCM to select ........................................................................................................................... 24

    6.1.1.6 The use of existing speciation computer codes in transport models ........................................................ 25

    7 CONCLUSIONS AND RECOMMENDATIONS ................................................................................. 26 8 REFERENCES ...................................................................................................................................... 27

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1 OBJECTIVES

    The present report aims to identify and evaluate the most current mathematical formulations describing adsorption-desorption processes including chemical exchange reactions in order to find the most suited model to use in contaminant transport models. In particular, the adsorption-desorption phenomena are discussed in order to 1) identify new or improved theoretical interpretations of surface phenomena and chemical equilibria, and 2) identify more advanced numerical procedures which already have been developed for application in contaminant transport models.

    2 INTRODUCTION

    For impact assessments of micropollutants in marine ecosystems, information on source terms including speciation, mobility and sediment interactions (influencing marine transport) as well as bioavailability (influencing biological uptake in marine organisms) is essential. For transport models, information on speciation and sediment interactions is vital as sediments can act as a sink, and contaminated sediments may also act as a diffuse source of contaminants in the future. As sensitivity analysis has shown, source term speciation and sediment interactions are key parameters contributing to the overall uncertainties in prognostic transport and impact assessment, implementation of more detailed process oriented codes should therefore be beneficial.

    Marine waters are dispersed multi-component systems where naturally occurring elements as well as artificially produced contaminants are present. Macrocomponents determine the physical-chemical properties of natural systems (e.g. pH, Eh, ionic strength) and information on macrocomponents in waters and sediments is needed for characterising the systems, when data on microcomponents is interpreted. For certain systems, especially oceanic waters, the concentration of macrocomponents remain quite stable, while in coastal areas or within estuaries, the concentrations may vary significantly due to run-off and river transport.

    For microcomponents like trace elements and radionuclides in marine waters, the concentrations vary extensively, reflecting the flux of matter from different sources (atmospheric transport, river transport, discharges to coastal areas) and distance from the source, transport processes and interactions within the water columns as well as removal processes like particle growth and sedimentation. Thus, the distribution of microcomponents in marine systems is strongly influenced by hydrological, geological and biological cycles and variable concentrations of microcomponents are especially pronounced in coastal areas and estuaries. As the concentrations usually are very low, three major issues should be focussed: 1. Trace chemistry phenomena which are negligible for macro-elements are dominant 2. Speciation of micropollutants is essential for mobility and biological uptake 3. Fully equilibrium can hardly be assumed.

    2.1 SPECIATION OF MICROPOLLUTANTS

    -5For micropollutants (concentration less than 10 M solution), trace chemistry phenomena

    occur; and reaction directions and mechanisms known from macrochemistry may change as microchemical processes, such as chemistry at phase boundaries, or chemistry of colloidal systems may become predominant. Furthermore, reaction rates may decrease as the probability of effective collisions decreases. Thus, relevant micro-chemical processes including sorption dynamics due to changes in pH, Eh, ionic strength and temperature as well

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    as colloidal transport and kinetics involved should be included in dispersion, transport or impact assessment models.

    In natural waters, most trace elements and radionuclides are present in different physico-chemical forms varying from low molecular mass (LMM) ions or molecules, via hydrolysis products and polymers to colloids and pseudo-colloids, or incorporated in inorganic or organic particles (Fig. 1). The borderline between categories is difficult to establish and transformation among categories occurs gradually. As natural waters are dynamic systems, the distribution of species exhibits spatial and temporal variations due to ongoing transformation processes. Owing to sorption to available surfaces, complexation with available inorganic as well as organic naturally occurring ligands, polymerisation and aggregation of colloids, LMM species are transformed to high molecular mass (HMM) species, while desorption, dissolution, displacement and dispersion processes may mobilise LMM species from surfaces of solids. Thus speciation models should not only focus on the distribution of species but also utilise kinetic information on transformation processes influencing the speciation. Information on kinetics and reversibility of transformation processes is, however, scarce in literature.

    Mobilisation mechanisms:Molecular Diameter

    desorptionmass (Da)

    dissolution

    dispersion

    (z-w)+z+, MLwSIMPLE IONSM(2-x)+(z-1)+, M(OH)xHYDROLYSIS M(OH)PRODUCTS

    (z-2u-v-w)+(OH)(L)(HO)uvw2nMONONUCLEARMOSPECIES

    (xz-2u-v-w)+O(OH)(L)(HO)xuvw2nPOLYNUCLEARM2SPECIES?10 1 nm

    

    COLLOIDS1-10 nm4 10 nmDiscrete phase?106 0.1 ?mPSEUDOCOLLOIDS10-450 nm?10

    

    PARTICLES450 nm 0.45 ?m

     1 ?m

    Molecular mass growth mechanisms:

    hydrolysis, complexation, polymerisation

    colloid formation, aggregation

    Figure 1. Size classes for the different physico-chemical forms of pollutants in aquatic systems. Transformation processes changing the distribution of species are indicated.

    2.2 SEDIMENT-SEAWATER INTERACTIONS

    Interactions taking place at phase boundaries, i.e. interaction of trace element and radionuclide species as well as organic components in water with sediments, occur via reversible and irreversible processes:

Processes Model Characteristics

    Physical sorption Consecutive layers Reversible, indifferent to charge

     Freundlich isotherm Unsaturated sites

     van der Waal attraction Mass attraction

    Electrostatic sorption Monolayer Reversible, indifferent to masses

     2

     Langmuir isotherm Saturated sites

     Coulomb attraction Opposite charge attraction

     pH sensitivity

     Ionic exchange

    Chemisorption Monolayer Irreversible, temp. dependent

     Langmuir isotherm Saturated sites

     Chemical bonds

     Surface complexation

     Fixation

    Therefore, the assumed interactions taking place at the seawater-sediment interfaces depend on the speciation of the element in question and the composition, structure and charge distribution of the surface sediment (sorbent). As most of the prevailing models include one or two of these interactions, assumptions are then automatically also made with respect to speciation of the interacting element, e.g., an ion exchange model is relevant for charged species only.

    H

    Electrostatic attraction:O2HNon-specifically sorbed ionO2

    Diffuse layerHO2O2H

    Electrostatic binding or ion pairing:HO2Outer sphere complex

    HOHO22Stern layerChemical bonding:HOO22HO2HInner sphere complex

    Negatively charged

    OOOOO

    SurfaceOOOOOO

    Figure 2. Schematic representation of mechanisms responsible for ion sorption on charged mineral surfaces (Dzombak & Hudson 1995).

    Assuming charged ions to be present in the water, different binding mechanisms can be plausible at the mineral-water interface (Fig. 2). For a metal-oxide surface being negatively charged a counterlayer of positively charged metal cations from the surrounding solution is formed due to electrostatic attraction of opposite charges, i.e., reversible electrostatic processes. Thus, a diffuse layer of non-specifically sorbed cations from the water adjacent to the oxide surface is formed. As indicated in Figure 2, some of the counterions may approach the surface more closely to form weak outer-sphere surface complexes (primarily through electrostatic binding or ion pairing) or stronger inner-sphere surface complexes (primarily through chemical bonding), i.e., irreversible chemisorption processes.

    The reversible and irreversible interactions of trace metals and radionuclides with surfaces in aqueous solutions can be distinguished using operational methods such as sequential extractions (Tessier et al. 1979). The metals extracted by different chemicals, can be

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    operationally categorised as reversibly (easily mobilised by inert electrolyte) or irreversibly (inert fractions mobilised by redox agents) associated with sediment components (Fig. 3). By performing such extractions at different contact times between sediment and trace-metal, apparent rate constants (k-k) for the interactions can be estimated between the operational 14

    phases.

    During recent years, major advances have been made in experimentally determining the properties of solid surfaces in contact with aqueous solutions. This is largely due to the development and application of new in situ experimental techniques. These techniques

    include X-ray absorption fine structure (XFAS), X-ray standing waves (XSW), infrared methods, second-harmonic generation (SHG) methods, surface X-ray diffraction and reflectivity as well as laser spectroscopy techniques (e.g. Brown et al. 1999).

    Water -Watersoluble species

    k12k

    k3

    EasilymobilisedInertfraction –k4fraction –SedimentIrreversibleReversibleinteractionsinteractions

    Figure 3. Box model dividing the sediment-water system into three operational compartments, which are connected by apparent rate constants, described by first order kinetics.

    Reversible surface interactions are usually rapid processes, while the diffusion through double layers into mineral lattices are very slow processes. Therefore, the assumption of equilibrium conditions is relevant for reversible processes (physical sorption, electrostatic sorption), but questionable for irreversible (chemisorption) processes. By adding radioactive tracers in cationic form to a seawater-sediment system, the equilibrium between species in solution and species easily mobilised from solid surfaces by an inert electrolyte (Fig. 3) is established between 2-7 days depending on temperature, while years are needed to establish equilibrium with the irreversible inert phases.

    Thus, the partitioning of trace elements, radionuclides and organic compounds among seawater and sediment components will be influenced by

    ? the speciation of trace metals and radionuclides in seawater,

    ? composition, structure and charge distribution on available sediment surfaces ? binding capacity of available sites

    ? the binding mechanisms involved influencing the kinetics

    ? competing element species

    which should be taken into consideration when models are selected for specific purposes.

    Particles with organic coatings are well known in aquatic ecosystems. For hydrophobic neutral organic compounds, physical sorption occurs and consecutive layers on particle

    surfaces can be formed. In addition, sorption is conceived as a solvent extraction process in which the solute molecule is preferentially ―dissolved‖ in the organic carbon component of

    the solid phase. This simplification has been applied for ―order of magnitude‖ predictions for the solid/liquid partitioning of many organic compounds under different conditions (Dzombak

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    et al. 1987). Because the interactions between solid and solute are assumed to be non-specific, sorption of hydrophobic trace organic compounds is roughly predicted from the information on the total organic carbon content of the sediment and from the octanol/water partition coefficient of the solute (octanol-water extraction). However, the uncertainties in these predictions is significant.

    2.3 KINETIC INFORMATION

    Developments within marine chemistry and radiochemistry have led to two strongly contrasting views of the same process: in marine chemistry the equilibrium relationships

    between chemical components is modelled, while within radiochemistry the isotopic disequilibrium has been utilised to develop kinetic models (Honeyman & Santschi 1992).

    Laboratory studies focussing the kinetic aspects of scavenging in marine systems were initiated by Nyffeler et al. (1984) and Li et al. (1984). Similar studies using radioactive tracers to obtain information on the interactions with marine sediments have also been performed recently (e.g. JRNC 1993, Oughton et al. 1997, Carroll et al. 1997). The data show that trace metals and radionuclides in cationic form exhibit a wide variation in the sediment/solution distribution coefficients, K, and that time is needed to reach equilibrium. However, Jannasch d

    et al. (1988) plotted the apparent sorption rate constants derived by Nyffeler et al. (1984) as a function of equilibrium K (Jannasch et al., 1988) and found a strong correlation. As most of d

    the kinetic data fit to first order kinetic expressions the interpretation of data with respect to distinguishing between underlying processes is complicated (Nkedi-Kizza et al. 1984). However, kinetic information is essential to reduce uncertainties in models predicting future transfer of microcomponents.

    As a major objective for this report concerns equilibrium conditions, models describing interacting processes at equilibrium are focussed. However, a final section on kinetic models has also been included as equilibrium approaches are questionable in dynamic systems. The discussion is also restricted to oxic environments where sorption is expected to be an important regulating mechanism, as opposed to anoxic environments where redox processes and precipitation/coprecipitation are likely to dominate. The partitioning of trace elements in oxic environments between solution and sediment components is described in terms of both empirical concepts (e.g. distribution coefficients), concepts related to electrostatic processes (e.g. ion exchange) and chemisorption processes (surface complexation).

    3 SORPTION OF CHARGED TRACE METALS AND RADIONUCLIDE SPECIES

    In principle two different approaches are used to describe the partitioning of microcomponents between solids and solutions at equilibrium conditions:

    1. Empirical partitioning relationships (such as K) i.e. conditional constants derived from d

    experimental field data or laboratory experiments.

    2. Conceptual models based on theoretical considerations describing interaction processes individually or in combination:

    a) physical sorption

    b) reversible electrostatic sorption

    c) irreversible chemisorption

    Because of the complexity of natural systems, the empirical approach has been widely used in describing the partitioning of solutes between mineral and water phases in geochemical applications, especially in transport models and engineering applications. Surface complexation models on the other hand, have been used primarily by aquatic scientists interested in developing a thermodynamic understanding of the coordinative properties of mineral surface ligand groups via laboratory investigations (Davis & Kent 1990).

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3.1 EMPIRICAL SORPTION MODELS

    3.1.1.1 Distribution coefficients

    Sorption is often described in terms of equations or partitioning relationships that relate the activity of a solute in water to the amount of the solute sorbed at constant temperature. The simplest of these expressions is the distribution coefficient (K) derived from the association d

    reaction:

    M ( SM (1)

    where M represents solute dissolved in the aqueous phase (e.g. ?g/l), and SM is solute M

    sorbed (e.g. ?g/kg). The K is usually defined as: d

    [SM] (2) Kd[M]

    The K is highly dependent on the conditions under which it is measured, e.g., pH, d

    temperature, background electrolyte composition and concentration, concentration of CO(aq), 2

    concentrations of competing adsorbates, etc. Thus, the distribution coefficient has little value in predicting the response of solute sorptive behaviour that may result from changes in either the aqueous or mineralogical composition of a system. Furthermore, the K approach gives no d

    information on binding mechanisms and equilibrium is assumed.

3.1.1.2 Langmuir isotherm

    The Langmuir adsorption model is applicable for electrostatic sorption and chemisorption reactions (monolayer model). The number of surface sites available for sorption is limited and must be specified at the outset. The surface reaction can be written:

    S + M ( SM (3)

    where M is an sorbing solute, S is an sorbing surface site, and SM represents the species of

    sorbed solute M. Assuming that all surface sites have the same affinity for solute M, the following mass law equation can be written:

    [SM]K (4) L[M][S]

    where K is the conditional Langmuir equilibrium constant. Making the further assumption L

    that the density of all adsorptive sites on the surface, S, is fixed and that M is the only T

    adsorbing solute, the mass balance law can be combined with the mass balance for surface sites:

     (5) SSMST

    to yield the expression commonly known as the Langmuir isotherm:

    ??K[M]LSMS (6) ??T1K[M]L??

    The K isotherm is an improvement over the K concept since the introduction of total surface Ld

    sites takes saturation effects into consideration. However, like the distribution coefficient equilibrium conditions are assumed and K is dependent on constant solution conditions and L

    is very sensitive to changes in pH. Thus, the application of K is questionable for variable L

    ecosystem conditions.

3.1.1.3 Freundlich isotherm

    In contrast to Langmuir monolayer model, the Freundlich isotherm is a consecutive layer model (unlimited sorption sites) and is useful for describing physical sorption. It is frequently observed that a plot of K versus SM results in a curve that is convex to the SM axis d

    instead of the straight line expected from the homogeneous site Langmuir expression. In these cases the data must be fitted to a multiple-site Langmuir expression or a generalised exponential isotherm, such as the Freundlich isotherm.

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The mass action equation representing the Freundlich model can be written:

    (1/n)M ( SM (7)

    [SM] (8) KF1/n[M]

    where 1/n is the mass action stoichiometric coefficient pertaining to M and K is the F

    conditional Freundlich stability constant. Like the K model, an unlimited supply of unreacted d

    sites is assumed. For the special case where n = 1, the Freundlich and activity K mass action d

    equations are identical. Thus, the fitting of experimental data to Langmuir or Freundlich isotherms elucidates binding mechanisms.

3.1.1.4 General partitioning equation +Because sorption of many ionic species is highly dependent on the concentration of H, it is

    important that sorption models are capable of predicting sorptive behaviour as a function of pH. The empirical relationships that relate sorption density to sorbate concentration are valid only for constant conditions, and thus, these models are generally insufficient for modelling processes in environmental systems. Honeyman and Leckie (1986) have considered a modified form of the distribution coefficient which describes the sorption of ion M in terms of the macroscopic observations of proton (or hydroxyl) exchange: χ[SM][H]K (9) part[M][S]

    where ~ is the apparent ratio of moles of protons released per mole of solute adsorbed. Despite the conditional nature of the constants, the approach has been useful in examining specific issues of adsorptive behaviour (Balistrieri & Murray 1983).

    3.2 SURFACE COMPLEXATION MODELS

    An alternative to the empirical modelling approaches are the surface complexation models (SCM), which extend the ion-association model of aqueous solution chemistry to include formation of chemical complexes on surfaces. SCMs treat surface functional groups as analogues of complexing ligands in solution. The main difference between the various models lies in the assumed structure of the surface complexes thought to be formed. While the models differ in their consideration of interfacial structure, all the models reduce to a set of simultaneous equations that can be solved numerically (Dzombak et al. 1987). These equations include: (1) mass law equations for all surface reactions under consideration, (2) a mole balance equation for surface sites, (3) an equation for computation of surface charge, and (4) a set of equations representing the constraints imposed by the model of interfacial structure. A number of different surface complexation models have been proposed during the last three decades. The models are distinguished by differences in their respective molecular hypotheses. Each model assumes a particular interfacial structure, resulting in the consideration of various kinds of surface reactions and electrostatic correction factors to mass law equations (Davis & Kent 1990).

    3.2.1 Interface (electrostatic sorption) models

    Four very common SCMs are the Constant Capacitance Model (CCM), the Diffuse Layer Model (DLM), the Basic Stern Model (BSM), and the Triple Layer Model (TLM):

3.2.1.1 Constant Capacitance Model (CCM)

    The constant capacitance model is a special form of the diffuse layer model (described below), applicable in theory only to high ionic strength and/or low potential systems. Assumptions:

    (i) all surface complexes are inner sphere complexes (chemisorption)

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    (ii) the constant ionic reference state determines the activity of the aqueous species and

    therefore no surface complexes are formed with ions from the background electrolyte (iii) one plane of charge represents the surface

    (iv) the relationship between surface charge density, and surface potential, is ??00

    CSa (10) σψ00F-22-1where C is the capacitance density (F m), S is the surface area (m g), a is the suspension -1-1-1density (g L), F is the Faraday constant (C mol), ? has units of mol L, and ? has units of cc

    V.

    3.2.1.2 Diffuse Layer Model (DLM) +In this model, all surface coordinating anions and cations are assigned to the same layer as H -and OH, and non-specifically adsorbed counterions are assigned to the diffuse layer. In the so called diffuse layer model, the relationship between surface charge and potential is fixed by electric double layer theory. The finite number of surface sites limits the value of the surface charge to reasonable values regardless of the ionic strength. Dzombak and Morel (1990) used two types of surface sites (high-affinity and low-affinity sites) to improve the DLM fit when varying pH.

    Assumptions

    (i) all surface complexes are inner sphere complexes

    (ii) no surface complexes are formed with ions in the background electrolyte (iii) two planes of charge represents the surface

    (iv) the relationships between surface charges and surface potentials are

    ? = ? (11) 0d

    Gouy-Chapman equation for symmetrical electrolytes

    ψFSa1/2dσDRTI (12) (8?)sinh()d0FRT2

    or general case

    1/2??Sa?? (13) ;;,:σsgn?2?DRTcexpzF?/RT1??iiddd0?F??i??

    where ? is the permittivity of vacuum, D is the dielectric constant of water, I is the ionic 0

    strength, sgn? = 1 if ? > 0 and sgn? = -1 if ? < 0 (where d represents the diffuse plane), dddd

    and c and z are the concentration and charge of solution species i. ii

3.2.1.3 Basic Stern Model (BSM) +-Two discrete planes of charge are assumed at the surface, with H and OH ions binding at the

    innermost plane and other specifically adsorbed ions binding at a second () plane separated

    from the first by a region of dielectric constant, ?. The solution side of the interface, which is 1

    assumed to begin immediately beyond the -plane of adsorption, is described with a Gouy-

    Chapman diffuse layer. A linear drop in potential between the planes of surface charge is assumed so that the two adsorption planes act as a parallel plate capacitor with a capacitance C. 1

3.2.1.4 Triple Layer Model (TLM)

    In this model, like the previous one, specific sorption of ions is assumed to take place in two +-separate planes (one for H and OH ions and one for other specifically sorbed ions) and a

    diffuse layer is assumed for the solution side of the interface. In contrast with the previous model, however, the diffuse layer is assumed to begin at the edge of a layer of bound water (dielectric constant ?), at some distance from the second adsorption plane. The triple layer 2

    model has the same number of fitting parameters as the Stern model, if the same kinds of

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