Review Paper Sorption Phenomena in Subsurface

REVIEW PAPER

SORPTION PHENOMENA IN SUBSURFACE SYSTEMS: CONCEPTS, MODELS AND EFFECTS ON CONTAMINANT FATE AND TRANSPORT*

g~ WALTER J. WEnER JR "~', PAUL M. MCGINLEY and LYNN E. KATZ ~) Environmental and Water Resources Engineering, The University of Michigan, Ann Arbor, MI 48109, U.S.A.

(First received March 1990; accepted in revised form September 1990)

Abstract--The behavior, transport and ultimate fate of contaminants in subsurface environments may be affected significantly by their participation in sorption reactions and related phenomena. The degree to which the resulting effects can be quantified and predicted depends upon the extent to which certain fundamental aspects of sorption are understood, and upon the accuracy with which these phenomena can be characterized and modeled in complex subsurface systems. Current levels of understanding of the reactions and processes comprising sorption phenomena are discussed in this paper, as are the forms and utilities of different models used to describe them. Emphasis is placed on concept development, on the translation of these concepts into functional models for characterizing sorption rates and equilibria, and on the application of these concepts and models for explaining contaminant behavior in subsurface systems. Examples are provided to illustrate the impacts of sorption phenomena on contaminant transport.

Key words--sorption, partitioning, soils, groundwater, subsurface systems, rate and equilibrium models, mass transfer, contaminant tranport

NOMENCLATURE k'= psuedo-first-order rote constant (I/t) K~ = ion-exchange selectivity coefficient between ions A A = area (L 2) and B a~ = chemical activity of species i k~ = overall mass transfer coefficient for diffusion into b -- Langmuir isotherm coefficient corresponding to the an adjacent solute accumulating phase (l/t) enthalpy of adsorption (L3/M) K D = distribution coefficient (L3/M) C~,~ = equilibrium solution phase concentration of species K{~ = distribution coefficient in phase j(L~/M) i (M/L 3) k¢ = effective mass transfer coefficient (L/t) C*=¢,1 equilibrium solution phase concentration of species K~ = equilibrium constant i in a singie-solute system at the same spreading K F = Freundlich isotherm constant [(L~/M) ~] pressure as that of a mixture (M/L ~) k F = forward rate constant (L3/M-t) C,.: -- concentration of species i associated with phase or kf = film transfer coefficient (L/t) interface j (M/L 3) Ko¢ -- organic carbon-normalized linear isotherm co- D, = effective diffusion coefficient (L2/t) efficient (L3/M) Dh = hydrodynamic dispersion coefficient (L2/t) Kow = octanol/water partition coefficient (dimensionless) Dh -- second-rank hydrodynamic dispersion tensor (L2/t) Kp--partition coefficient (dimensionsless) D~,a = apparent dispersion coefficient (L2/t) k R = reverse rate constant (l/t) D1 = free liquid diffusion coefficient (L2/t) m = Morse potential constant (l/L) D m = minimum energy in the Morse potential model N = total number of species in solution (ML2/t 2) n = Freundlich isotherm exponent (dimensionless) E=Coulombic constant of proportionality (ML3/ F-Q 3) %OC = percent soil organic carbon content q = mass of solute sorbed per unit mass of sorbent o = minimum energy level for attractive forces in (M/M) Lennard-Jones potential (ML2/t 2) q, -- mass of solute sorbed per unit mass of sorbent at f, j = activity coefficient of species i in phase j radial distance r (M/M) F~,'~ = flux of species i in direction x (M/L2-t) qo,~ = mass of solute i sorbed per unit mass of sorbent at HSA =hydrocarbonaceous molecular surface area (L 2) equilibrium (M/M) k = Boltzman's constant 1,38 x 10-~s erg/degree q~ = solid-phase concentrations of species i in single- Kelvin (ML:/F-T) solute system with the same spreading pressure as that of the mixture (M/M) *This paper was the basis of the 1990 Distinguished Lecture q~.j = solid-phase concentration of species i associated by Walter J. Weber Jr for the Association of Environ- with phase or interface j (M/M) mental Engineering Professors. Q = charge

499 500 WALTER J. WEBER JR et al.

Q0 = Langmuir isotherm coetficient corresponding to the sorption occurs and set of local conditions yields a mass of sorbate per unit mass of sorbent at mono- unique mass distribution, which may be either stable layer coverage (M/M) qr = total mass of solutes sorbed per unit mass of or transient. Thermodynamic considerations govern sorbent (M/M) contaminant mass distributions ultimately achiev- R = gas constant 8.314 J/K-mol = 8.314 x 107 erg/ able, and thus "stable". The rates at which such K-tool (L2/t 2-T) distributions are approached involve separate but r = radial distance (L) re = Morse potential minimum separation distance (L) equally important considerations, considerations Rr = retardation factor (dimensionless) which frequently determine the relative significance of Rp = particle radius (L) sorption with respect to other reaction and transport T = absolute temperature degrees Kelvin processes operating in subsurface environments. Pre- t = time (t) diction of contaminant fate and transport requires V = volume (L 3) ¥ = volume of solute-accumulating phase adjacent to a characterization and quantification of both energy mass transfer domain (L 3) balances and rates associated with sorption processes. v = reaction velocity (M/L3-t) State-of-the-art "know-how" in this regard in- v = the pore-velocity vector (L/t) cludes a reasonable understanding of parameters Vm,/= the molar volume of phase j (L 3) v~= one dimensional fluid-phase pore (interstitial) which influence rates and extent of reactions, an velocity in the z direction (L/t) extensive base of empirical observations, and a com- xi,j = the mole fraction of solute i in phase j (dimension- prehensive set of descriptive and predictive models less) predicated on various mechanistic representations of z¢.i = the mole fraction of solute i in the sorbed phase sorption phenomena. This paper explores our current (dimensionless) z~ = charge species i(Q) level of understanding of sorption processes, and the = empirical coefficient relating sorption to the co- form and utility of different models used to describe solvent content of the solvent (dimensionless) them and their effects. Particular emphasis is given /t c = volume fraction of a cosolvent in an aqueous phase to concept development, and the extent to which (dimensionless) = surface tension (M/t 2) such concepts can be applied in understanding A),' = difference in HSA/solvent interfacial tensions and explaining contaminant behavior in subsurface (M/t 2) systems. E = porosity, void volume per unit total volume (dimensionless) q~o~= mass of organic carbon per unit mass of solid 2. SORPTION PHENOMENA (dimensionless) OSM = mass of solid phase associated with the mobile fluid Sorption interactions generally operate among all per unit total mass of solid (dimensionless) phases present in any subsurface system and at the p~ = density of phase i (M/L 3) interfaces between these phases. Solutes which 0/= volume of phase or region j per unit total volume (dimensionless) undergo sorption are commonly termed sorbates, the ~kc = Coulombic potential energy (ML2/t2-Q) sorbing phase the sorbent, and the primary phase ~ku = Lennard-Jones potential energy (ML2/t 2) from which sorption occurs the solution or solvent. ~kM = Morse potential energy (ML2/t 2) Two broad categories of sorption phenomena, ad- ~b°=electrochemical potential at the near approach sorption and absorption, can be differentiated by the plane in a surface complexation model (ML2/t2-Q) degree to which the sorbate molecule interacts with a = electrochemical potential at the fl plane in a surface and is free to migrate between the sorbent phase. complexation model (ML2/t2-Q) In adsorption, solute accumulation is generally re- a = parameter in the Lennard-Jones potential relation- stricted to a surface or interface between the solution ship (L) F = surface excess (M/L 2) and adsorbent. In contrast absorption is a process in /-/~ = spreading pressure of species i (M/t 2) which solute transferred from one phase to another #~ = chemical potential of species i (ML2/t 2) interpenetrates the sorbent phase by at least several /~0 = standard chemical potential of species i (ML2/t2). nanometers. An additional variation of the process occurs if a sufficiently high accumulation of solute l. INTRODUCTION occurs at the interface to form a precipitate or some other type of molecular solute-solute association, e.g. Sorption processes involve an array of phenomena a polymer or a micelle. Such processes differ from which can alter the distribution of contaminants both adsorption and absorption in that they result in between and among the constituent phases and inter- formation of new and distinct three-dimensional faces of subsurface systems. The interchanges of mass phases. Because adsorption processes generally yield associated with such processes impact the fate and surface or interface concentrations of solute greater transport of many inorganic and organic substances. than those in the bulk phase, it is possible for The effects can be complex, given the diversity, precipitation or association to occur on a surface in magnitude and activity of chemical species, phases the absence of a solution phase reaction of the same and interfaces commonly present in contaminated type. Although such association reactions are often subsurface environments. Each combination of classified as separate processes, they must in fact be solute, sorbing phase, interface, phase from which preceded by adsorption or absorption. Sorption phenomena 501

Sorption results from a variety of different types of referred to as polar molecules. Interactions between attractive forces between solute molecules, solvent polar molecules or between polar molecules and non- molecules and the molecules of a sorbent. Such forces polar molecules--in which dipole moments are thereby usually act in concert, but one type or another is just induced--represent one class of "physical" sorption. as usually more significant than the others in any In most cases, such interactions are short range, the particular situation. Absorption processes, such as associated energy decreasing inversely with the sixth dissolution of a relatively immiscible phase into an power of the distance separating the interacting mol- aqueous phase or accumulation of a lipophilic sub- ecules. There are cases, however, of longer range stance in an organic phase, involve exchanges of permanent dipole interactions in which the potential molecular environments. In such cases, the energy of energy is inversely related to the third power of the an individual molecule is altered by its interactions distance between molecules; for example polar mol- with the solvent and sorbent phases. The distribution ecules which are specifically oriented (Hiemenz, 1986). of the solute between phases results from its relative A more general class of physical sorption is associ- affinity for each phase, which in turn relates to the ated with forces attributable to rapidly fluctuating or nature of the forces which exist between molecules of instantaneous dipole moments resulting from the the sorbate and those of the solvent and sorbent motion of electrons in their orbitals; so-called phases. These forces can be likened to forces which London dispersion forces. Energies associated with arise in classical chemical reactions. Adsorption also interactions of this type also decrease inversely with entails intermolecular forces, but in this case it is the sixth power of the distance between molecules. molecules at the surface of the sorbent rather than The magnitude of physical sorption forces can be bulk phase molecules which are involved, and the estimated from measurements of differential heats of former typically manifest a broader range of inter- adsorption. Values for interactions of the London- actions. Accordingly, three loosely defined categories dispersion type for small molecules are generally of of adsorption--physical, chemical and electrostatic-- the order of l-2kcal/mol (Kiselev, 1965). More are traditionally distinguished, according to the class specific interactions have higher heats of sorption. of attractive force which predominates. Some signifi- The relatively weak bonding forces associated with cant features of these different interactions and physical sorption are often amplified in the case of classes of adsorption are summarized in Table 1. hydrophobic (more generally, solvophobic) molecules Forces associated with interactions between the by substantial thermodynamic gradients for repulsion dipole moments of sorbate and sorbent molecules from the solution in which they are dissolved. commonly underlie physical sorption processes. Although the sorption bond may still be attributable Dipole moments arise from charge separation within to dispersion-type interactions, the combined effect in a molecule and can be either permanent or induced. aqueous system is often referred to as "hydrophobic Molecules possessing a permanent dipole moment are bonding" (Haymaker and Thompson, 1972).

Table 1. Characteristic interactions associated with categories of adsorption

Category and Representation of Interaction Characteristic Interaction Interaction Range CHEMICAL @ ~ H O Short Range Covaient H 2 2

/o Short Range

ELECTROSTATIC Ion-Ion O r O 1#

Ion-Dipole / r (~__ 1#2 J PHYSICAL ¢ 1# 3 Dipole-Dipole --/--'" (Coulombic) / f

(Keesom energy) ~ A,A~/ l/r 6

Dipole-InducedDipole ,/( r A (Debye energy) ~) ,~¢ 1/r 6

Instantaneous Dipole-InducedDipole ,~ r 1/r 6 (London dispersion energy) Ira, w *Adapted from lsraelachvili, 1985. 502 WALTERJ. WEBERJR et al.

The existence of an energy acting to drive hydro- the Lennard-Jones relationship, its graphical form phobic molecules out of aqueous solution can be will be similar. understood in the context of the structure of water. Forces of greater intensity and longer range exist It is commonly envisioned that water molecules exist between discretely charged entities. These forces de- in one of two principal types of structural associ- rive from specific electrostatic interactions between ations in aqueous phase. The first is one in which each localized charges, and exhibit much higher heats of water molecule is tetrahedrally coordinated to four sorption than those associated with physical sorption. others via hydrogen bonding, yielding a structure Electrostatic forces extend over long distances, vary- similar to that of crystalline ice. The second is an ing inversely with the square of the distance between agglomeration of more densely packed but less well molecules and directly with the product of the ordered molecules. A relatively non-polar dissolved charges. The relationship between potential energy solute molecule is held in aqueous solution by an and distance between charges is defined by arrangement of the ice-like water molecules, and the Coulomb's law: dissolution reaction is generally exothermic. This 2122 favorable enthalpy of solution is countered, however, ~Oc = E--. (2) by an unfavorable entropy resulting from the in- r creased ordering of solvent molecules (Nemethy and The magnitudes of the charges associated with each Scheraga, 1962; Haymaker and Thompson, 1972). of two interacting species are represented by zl and Solute molecules can thus be driven from solution at z2, respectively, and E is a constant dependent on the concentrations well below maximum solubility, that properties of the solvent. As shown by equation (2), is at levels below those at which they could precipi- these forces can be attractive in the case of oppositely tate, if the system can achieve a state that is thermo- charged species or repulsive between those having like dynamically favorable to sorption. Chlorinated charges. hydrocarbons, for example, tend to sorb readily The final category of sorption defined according to to organic-rich soils because hydrocarbon-natural predominant surface-solute interaction is chemical organic interactions are energetically preferred to sorption or chemisorption. The bonds that form hydrocarbon-water interactions. between solute molecules and specific surface chemi- To this point the discussion has focused primarily cal groups in this type of sorption have all of the on attractive forces. At near approach, or small characteristics of true chemical bonds and are charac- intermolecular separation, repulsive forces may well terized by relatively large heats of sorption. The dominate. Repulsive forces--variously referred to as reactions may involve substantial activation energies exchange, hard core or Born repulsion forces-- and be favored by high temperatures. Chemical become negligible beyond an extremely small and bonding between a sorbate molecule and a sorbent characteristic intermolecular distance referred to as site can be represented in terms of the Morse poten- the van der Waals radius. Empirical descriptions of tial energy relationship, which was developed for a repulsive forces have generally employed relation- covalent bond between two identical molecules, and ships in which the energy varies inversely with the has the functional form (Gasser, 1985): twelfth power of distance (Israelachvili, 1985). Sum- ~M = Dm[1 -- e--mtr--re)] 2" (3) mation of the attractive and repulsive forces leads to a total intermolecular pair potential relationship The term D m in equation (3) is the minimum energy, known, in the case of dispersion forces, as the r e is the equilibrium separation of the molecules and Lennard-Jones or "6-12" potential: m is a constant. The variation of the Morse potential energy with separation distance exhibits a form simi- ~/LJ ~-- 4e[(a/r) l: - (o'/r)6] • (1) lar to that of the Lennard-Jones potential, although The term ffLJ represents the potential energy between the former yields a much greater minimum energy a pair of molecules separated by a distance of 2r, e and a smaller separation between molecule and the minimum energy level for which attractive forces surface sites at the minimum. correspond to negative potential, and tr the half- Categorization of the primary classes of inter- distance between molecules when ~bLj= 0. As noted actions leading to sorption reaction provides a useful previously, the potential energy between molecules means to bridge the gap between detailed enumer- attributable to other types of physical interactions ations of intermolecular forces and working descrip- may vary in magnitude, but the general shape of the tions of observed sorption phenomena. In reality, potential energy relationship remains the same. however, such forces do not act independently; it is, Moreover, extension of the Lennard-Jones potential, rather, the effect of their action collectively and in which was derived for interactions between two concert which dictates the sorptive separation of molecules, to the interaction of a molecule and a substances in any system. As such, there are two surface necessitates summing the pairwise inter- predominant means by which the sorption is motiv- actions between the molecule and surface atoms. ated. The first derives from the net forces of affinity Although the resulting relationship will obviously between the sorbate and the sorbent, here defined have different values for the exponents than those of as "sorbent-motivated" sorption. Cation exchange Sorption phenomena 503 reactions with clays, for example, are sorbent- compounds it is common to take the pure solute as motivated sorptions in which the electrostatic charge the reference state with correction for pure solutes characteristics of the sorbent are overwhelmingly which are solid at the temperature of interest attractive to the sorbate. The second involves the sum (Yalkowsky and Valvani, 1980). For purposes of this of adverse interactions of the sorbate with its solution discussion, however, the activity coefficient may be phase, yielding "solvent-motivated" sorptions. The taken to reflect the extent to which intermolecular repulsion by water of an oil or other hydrophobic interactions in the solution (solute-solvent and contaminant leading to its accumulation at a solute-solute) differ from those in the standard state soil-water interface, which may or may not be in- (solute-solute) (Karickhoff, 1984). different to the sorbate, constitutes a principally The partitioning of a solute between two phases solvent-motivated sorption. As expected, the com- can be expressed in terms of the ratio of its respective bined influence of all intermolecular forces in com- concentrations, Ct, in each phase. Assuming the same plex systems usually leads to sorption processes standard state for i in both the solvent (1) and sorbent which are not as easily categorized as specifically (2) phases and low concentrations of solute in each or exclusively either sorbent motivated or solvent phase, this ratio has a constant value given by: motivated, but rather which fall somewhere in the Ct,._.2_2=Vm, lXi, 2 Vm, I f/, 1 interlying continuum. = K~. (7) Ct,! Vm, 2Xt, 2 Vm, 2ft, 2 3. SORPTION EQUILIBRIA The term Vm is the molar volume of each phase, and Kp is termed the phase "partitioning" coefficient for 3.1. Absorption processes the solute. Equation (7) relates the distribution of The eventual equilibrium distribution of contami- solute between the phases to the ratio of the respect- nant mass between solution phase and a sorbent ive activity coefficients, reflecting the relative inter- or interface is dictated by a corresponding energy molecular interactions of the solute in those two balance, and thus may be categorized thermo- phases. In aqueous phases, the hydrophobic inter- dynamically. This characterization is somewhat more actions described earlier coupled with the choice of obvious for absorption than for adsorption, in that reference state can lead to activity coefficients much the former relates more to issues of classical chem- greater than unity, a reflection of the significant istry and energy balances within and between discrete difference in interactions which arise between the phases, whereas the latter often involves ill-defined solute and the aqueous solvent and the solute in the surfaces and interfaces. pure solute reference state. The differences in molecular environments of a contaminant in aqueous and sorbent phases which 3.2. Adsorption processes manifest themselves in absorption processes can be The properties of a system most significant for this described using classical thermodynamics. At equi- type of sorption process are those related to the librium, there is no driving force for further net surface at which accumulation occurs. Driving forces chemical change and the chemical potential,/~t, of a for attainment of chemical equilibrium in homo- solute i must be equal in the solvent and sorbent geneous phases relate to reduction of the free energies phases: of bulk systems, whereas the driving force for a surface reaction is a reduction in surface energy. One & so~vent = #t, so~'~n~" (4) common manifestation of surface energy occurs when The chemical potential is a function of the chemical a drop of liquid placed on a flat solid surface resists activity, at, and the standard state chemical potential, spreading, and attempts to gain or retain a nearly p0, of species i: spherical shape. The liquid in this case is attempting to minimize its free surface energy. This phenom- #, _ #o + RT In at. (5) enon, the development of a tension at the surface, Chemical activity is dimensionless, but its value de- results from attractive forces between molecules of pends on the units of concentration and on the choice the solid plate or molecules of liquid within the drop. of a standard state. The activity of a solute can be Because molecules of the liquid are attracted more related to its mole fraction, x t, through an activity strongly to one another by cohesion forces than they coefficient, f: are to molecules of the solid plate at the liquid-solid interface, molecules of liquid at the surface will be a, =f,&. (6) pulled mainly toward the interior of the drop. The The activity coefficient is defined with respect to a same effect applies to the air-liquid interface, which reference state. Two reference states are common: the thus tends to decrease its exposed surface as a result infinitely dilute state, where ft approaches unity as xt of the surface tension developed by the intradrop approaches 0, and the pure solute state, where ft molecular cohesion forces. This imbalance of inter- approaches unity as xt approaches 1. The best choice molecular forces is illustrated schematically in Fig. 1. for the reference state depends on the system to be A pure liquid always tends to reduce its free surface examined. For liquid-liquid partitioning of organic energy through the action of surface tension. From a 504 WALTERJ. WEBERJR et al.

Fig. 1. Illustration of surface tension arising from the imbalance of molecular attraction forces in a liquid drop (adapted from Weber, 1972). molecular point of view, enlarging a surface requires Equation (9) states that any solute which reduces the the breaking of bonds between surface molecules of interfacial tension (i.e. dy/dC < 0) results in an in- the liquid phase, and the forming of bonds between crease in F and thus will be present at the interface these molecules and those of adjacent phases. in higher concentration than in bulk solution. For A wide variety of soluble substances can alter the d?/dC > 0, negative adsorption is experienced. Water surface tension of a liquid. Surfactants, for example, has a relatively high surface tension (73 dynes/cm for lower the surface tension of water and cause it to a pure water-air interface at 20°C; Weber, 1972) spread on a solid surface, resulting in a wetting of which can be reduced by the presence of other solutes. that surface. Such substances are therefore termed Typical changes in surface tension at water-vapor wetting agents. If a surface active solute is present in interfaces are depicted in Fig. 2 for three classes a liquid system, a decrease in the surface tension of solutes, an inorganic electrolyte (e.g. a calcium of that liquid will occur upon movement of the chloride salt), a relatively simple organic chemical substance to the surface. Stated another way, any such as a chlorinated ethene (e.g. trichloroethylene) solute known to lower the surface tension of a liquid and a surface active organic compound (e.g. an alkyl will migrate to, and adsorb at, the interface of that sulfate). It is apparent that both organic solutes act liquid with some other phase. to decrease surface tension and tend to adsorb at the It is possible to define rigorous relationships for water-vapor interface, although, just as apparently, conditions in which equilibrium interfaciat tension is to different degrees. In contrast, the inorganic electro- reduced with increasing concentration of a solute lyte increases the surface tension and tends to migrate sorbed at an interface. Such relationships are away from the water-vapor interface. rooted in the fundamental expression developed by J. Willard Gibbs in 1878 for relating a change in the interfacial tension, ?, at the surface of a phase to the adsorption of solutes at the surface. If Fi is defined as the equilibrium amount of solute, i, adsorbed per unit area of surface in excess of its concentration in the SimpleElectrolyte bulk solution (i.e. the "surface excess"), the Gibbs )t mnm~mmBBm~mn equation at constant temperature and pressure has the form (Weber, 1972):

d~ = - ~ F id#i. (8) i=1 ~ AmphipathicSolute For dilute solutions of one solute of concentration e C, the Gibbs equation may be approximated by: Fig. 2. Examples of variations in surface tension at water-air interfaces with changes in the aqueous phase dC d? = - Rrr --. (9) concentrations of different solutes (adapted from Hiemenz, c 1986). Sorption phenomena 505

4. EQUILIBRIUM MODELS 20 4.1. Phenomenological models 15 Models for characterizing the equilibrium distribution of a solute among the phases and interfaces of an environmental system typically relate the amount .~_ 10 of solute, qe, sorbed per unit of sorbing phase or Q; interface to the amount of solute, C,, retained in the cr solvent phase. An expression of this type evaluated at a fixed system temperature constitutes what is termed a sorption "isotherm". Examples of phenomenologically different equilibrium patterns are illustrated graphically in Fig. 3. 0 x"2000 4000 6000 8000 Ce A number of conceptual and empirical models have (p.gIk ) been developed to describe these various adsorp- Fig. 4. Linear isotherm model fit to experimental equi- tion patterns. The most simple is the linear model, librium data for sorption of tetrachloroethylene (TTCE) by which describes the accumulation of solute by the Ann Arbor II soil in CMBR systems. sorbent as directly proportional to the solution phase concentration: and transport because they substantially reduce the mathematical complexity of the modeling effort. qe = KDCe. (10) Even when a particular set of data are reasonably well The constant of proportionality or distribution co- described by a linear model, however, caution should efficient, Ko, is often referred to as a partition be exercised in application of the model because it coefficient. However, Ce and qo are typically expressed may not be valid over concentration ranges beyond in terms of mass per unit volume and mass per those represented by the data to which it is calibrated. unit mass, respectively, and thus KD, unlike the Kp The Langmuir model, perhaps conceptually the presented earlier, is not dimensionless. most straightforward non-linear isotherm model, was An example of a linear isotherm is depicted in developed originally for systems in which sorption Fig. 4 for a particular set of experimental equilibrium leads to the deposition of a single layer of solute data obtained in our laboratories for sorption of molecules on the surface of a sorbent. This model is tetrachloroethylene (TTCE), a moderately hydro- predicated on the assumptions that the energy of phobic, volatile compound, by a moderately low sorption for each molecule is the same and indepen- organic content soil, Ann Arbor II (%OC = 0.5), in dent of surface coverage, and that sorption occurs completely mixed batch reactors (CMBRs). The lin- only on localized sites and involves no interactions ear isotherm is appropriate for sorption relationships between sorbed molecules. Given these assumptions, in which the energetics of sorption are uniform with the Langmuir model can be derived variously by increasing concentration and the loading of the sor- mass action, kinetic, or statistical thermodynamic bent is low ("Henry's region" sorption). It accurately approaches. The resulting expression is in each case: describes absorption and has been found to ade- Q°bC~ quately describe adsorption in certain instances, most qe= 1 +bC~ (ll) commonly at very low solute concentrations and for solids of low sorption potential. When justified, linear The parameter Q0 represents the sorbed solute con- approximations to sorption equilibrium data are centration on the sorbent corresponding to complete particularly useful in modeling contaminant fate monolayer coverage, i.e. a "limiting capacity", and b is a sorption coefficient related to the enthalpy of adsorption. At low surface coverage the Langmuir Favorable isotherm reduces to a linear relationship. Calibration of the model to a set of experimental data can be accomplished either by nonlinear regression or by simple regression of a linearized form of the model, the most common of which results from inversion of ~" /Linear Adsorption ~. equation (11): 1 i 1 1 = ~-~ bQ o Ce. (12) An example of an application of equation (12) to description of experimental CMBR equilibrium data obtained in our laboratories for sorption of trichloro- Aqueous Phase EquilibriumConcentration, Ce benzene (TCB) by Ann Arbor I soil (%OC = 1.14) is Fig. 3. Illustration of general types of sorption isotherms. presented in Fig. 5. Note by comparison of Figs 4 and 506 WALTER J, WEBER JR et al.

5 that the soil of higher organic content and higher 1000 sorptive potential (Ann Arbor I) yields a distinctly non-linear sorption pattern, while that of lower organic content and lower sorptive potential (Ann 100 Arbor II) yields a pattern which is reasonably described by a linear isotherm model. 8. lo The Freundlich isotherm is perhaps the most widely used nonlinear sorption equilibrium model.

Although both its origins and applications are for the 1.0 most part empirical, the model can be shown to be thermodynamically rigorous for special cases of sorption on heterogeneous surfaces. This model has the 0.1 ...... general form: 1.o lo loo looo lo#oo Ce q~= KvC ~. (13) Fig. 6. Freundlich isotherm model fit to experimental equi- The parameter KF relates to sorption capacity and n librium data for sorption of trichlorobenzene (TCB) by to sorption intensity. For determination of these Wagner soil in CMBR systems. empirically derived coefficients, data are usually fit to the logarithmic form of equation (13): The lAST model has theoretical roots in the Gibbs log qo = log K r + n log Ce. (14) equation, and thus provides a useful thermodynamic An example of such a fit is presented in Fig. 6 for approach to description of multisolute sorption be- equilibrium data we have obtained for sorption of tri- havior. The IAST model can be used to solve for the chlorobenzene (TCB) on Wagner soil (%OC = 1.2) In adsorbed quantities, qo, i, of each of a mixture of CMBR systems. It is not uncommon for the Langmuir solutes, requiring only the corresponding single solute and Freundlich models to have roughly equal utility isotherm relationships for each solute and sorbent of interest. The solution method described here sum- for describing nonlinear sorption phenomena over moderate ranges of solution concentration, but major marizes the numerical aspects; a more rigorous expo- sition of the theory is available elsewhere (Radke and differences between the models are usually apparent Prausnitz, 1972). The reduction in surface tension over wide ranges and high levels of concentration. which accompanies adsorption is commonly termed In multiple solute systems competition between the spreading pressure,/7. IAST equates the spread- solutes for sorption may occur as a result either of ing pressure of each component,//i, with that of the differences in sorption energies (sorbent or solute system at equilibrium. The spreading pressure of a heterogeneity) or because of site limitations. Models solute is computed from single solute isotherm data based on adaptations of the Langmuir and through integration of q~,/Ce, i with respect to C,,~ Freundlich isotherms have been used to describe over the range from 0 to that concentration value, multi-solute sorption, but these adaptations are re- C* corresponding to the concentration required in stricted in their applicability by the limiting assump- ¢,t, a single-solute system to yield the same spreading tions of the elementary models. More accurate pressure as the mixture, that is: characterization of multicomponent adsorption equilibria is often provided by the ideal adsorbed solution _ RT (ct,,q~ ,' .~,., theory (IAST) model. /7 =//~ = -- t .... uLo i. (15) A J0 Co,, ' 0,20 ..... ill When the equilibrium relationship between C~ and q~ can be expressed in a functional form such as one of the isotherm models discussed above, equation (15) 0.15 becomes an analytic expression for the spreading pressure of each solute as a function of its solution phase concentration or, alternatively, the concen- 0.10 tration of each solute as a function of.the spreading pressure of the system. The IAST model assumes an 0,05 ideal adsorbed phase (i.e. infinitely dilute sorbed phase) where the mole fraction, ze,~, of each solute in the sorbed phase is equal to the ratio of the solute 0.00 concentration in the mixture, Ce,. to its concen- 0.00 0.02 0.04 0.06 0.08 0.10 tration, Co*. in single-solute solution having the same 1/Ce spreading pressure as that of the mixture: Fig. 5. Langmuir isotherm model fit to experimental equilibrium data for sorption of trichlorobenzene (TCB) by Ann C~,, (16) Ze'i-'~" C*." Arbor I soil in CMBR systems. ¢, 1 Sorption phenomena 507

The mole fraction of sorbed solute is also equal to the verification of the concept or mechanism upon which ratio of the sorbed amount, q~,~, of i to the total it is based• The ability of any phenomenological sorbed phase concentration of all species, qe, r: model to describe observed data may establish its utility for a specific set of conditions, but the inherent ze. ~= --.qe, i (17) lack of mechanistic rigor associated with such models qe, r dictates against extrapolation to ranges of system The use of a set of solution phase solute concen- conditions not experimentally quantified. Indeed, trations to solve for the individual sorbed phase none of the isotherm models discussed here or other- solute concentrations requires solving for the spread- wise available has been demonstrated to be capable ing pressure of the system (/7), the sorbed phase mole of describing data over a wide range of conditions fractions of each solute (ze.~) and the single solute without parameter recalibration. It is not uncommon solution phase concentrations for each solute at for a model to describe observed sorption behavior the system spreading pressure (C*~). The above for a given sorbate/sorbent combination under one equations can be solved simultaneously for these set of conditions but fail to do so when the system unknowns using a variety of techniques (see, for conditions change. example, Weber and Smith, 1988). Once the spreading pressure of the system and the 4.2. Mechanistic models sorbed phase mole fraction of each solute is known, Models providing "first principle" description of the sorbed phase concentration of each solute is given the energetics of intermolecular reactions underlying by: sorption have also been developed. These models include mechanistic characterization of the ion ex- qe. i = qe. rZe. i. (18) change, surface complexation and hydrophobic sorp- The total sorbed phase concentration of all species, tion reactions discussed at the outset of this paper. qe, r, is calculated from: Such models can often provide insights into mechanisms controlling sorption reactions in particular types 1 ~ ze'i. (19) of systems, and thus aid in the analysis of anticipated system responses to changes in critical conditions• The single-sorption value corresponding to the same spreading pressure as that of the multi-solute sol- 4.2.1. Ion exchange and surface complexation ution, q**, can be obtained from the isotherm- Soil materials typically contain a variety of surfaces spreading pressure relationship. which exhibit electrical charge characteristics, which An example application of the lAST model in turn can exert strong influence on the sorption of to equilibrium data we have obtained for sorption of ionic and polar species. Such surface charges are TTCE and DCB from a bisolute mixture in CMBR instrumental, for example, in the sorption of metal systems is illustrated in Fig. 7. species in subsurface systems. The charge on the It is important to note that the assumptions associ- surface must be counterbalanced in the aqueous ated with the conceptual developments of the fore- phase to maintain electroneutrality. As a result, an going isotherm models are rarely satisfied in natural electrical double layer exists at interfaces• This double systems. Thus the fact that any one of them may layer consists of the charged surface sites and an provide a phenomenological description of a sorption equivalent aqueous-phase excess of ions of opposite process in any given situation should not be taken as charge (counter-ions) which accumulate in the water near the surface of the particle. The counter-ions are 1130. attracted electrostatically to the interfacial region, o giving rise to a concentration gradient, which in turn o [] sets up a potential for random diffusion of ions away 75. t~ from the surface. The competing processes of electro- D D/O g aa / o static attraction and counter-diffusion spread the ~[] O0 charge over a diffuse layer in which the excess

[] u 0 concentration of counter-ions is highest immediately adjacent to the surface of the particle and decreases gradually with increasing distance from the solid- 25. ~ aa ~. O~150 I [] Single.Solute Data ' water interface. Such distributions are illustrated schematically in Fig. 8. [ ~ Ideal Adeorbcd Solution Theory Several types of reactions can be attributed to 0 ! 0 1000 2000 3000 4000 forces associated with charged sites. Ion exchange Ce (~g/L) reactions resulting from the action of electrostatic forces occur at fixed sites on soil surfaces (Stumm and Fig. 7. Single-solute and bi-solute equilibrium data and lAST model prediction for sorption of tetrachloroethylene Morgan, 1981; James and Parks, 1982). Fixed-charge on Ann Arbor I soil in the presence of dichlorobenzene in sites, those not subject to change in solution phase CMBR systems. concentration, result from isomorphic substitution of 508 WALTER.J. WEBERJR et al.

Stern Diffuse Bulk Surface Layer Layer Solution deprotonation reactions, and are thus highly pH dependent. Carboxyl and phenolic hydroxyl groups are the primary surface functional groups involved in surface complexation reactions on soil organic matter. The most abundant surface functional group i e®• ® ® participating in surface complexation reactions on oxide surfaces and clay minerals is the hydroxyl group, which is amphoteric and extremely reactive (Sposito, 1984a). It is the reactivity of such sites which induces a strong pH dependence for sorption ®® @ of metal ions by natural solids. @® ® A number of surface complexation models have been developed over the past several decades. These utilize mass law relationships and mass and charge Fig. 8. Schematic representation of an electrical double layer. balance equations to describe equilibria between solution species and surface complexes, and various ions in the lattice structure of clay-like minerals. A hypotheses regarding the structure of the interfacial number of relationships have been developed, to region to identify the location of surface complexes describe ion-exchange equilibria, including equations and describe the diffuse layer-charge potential based on the Guoy-Chapman model for the diffuse relationship. General reviews are available in the double layer (Eriksson, 1952). The most common literature (Sposito, 1984b; Barrow and Bowden, description is made by analogy to a chemical reac- 1987). For purposes of illustration one such tion. For example, the exchange of a cation A "+ of model, the Triple Layer Model (TLM), is briefly charge n, dissolved in solution, for a monovalent summarized here. cation B +, associated with an adsorbent surface, can The triple layer model, in its current version (Hayes be written in terms of simple stoichiometry for a and Leckie, 1987), assumes that three planes of fixed-charge site, S-, as: adsorption exist within the electrical double layer and that the charge at each plane results from surface A "+ + n(S-)B + = (S-).A "+ + nB +. (20) complexation of ionic solutes. A schematic representation of the interracial structure of the model is Continuing the analogy, a parameter corresponding presented in Fig. 9. The a ° and a' planes shown in in form to the mass law equilibrium constant but this figure are generally referred to as the inner and more correctly referred to as a "selectivity coefficient" outer spheres, respectively. Metal ions are presumed can be defined in terms of the chemical activities of to undergo either inner sphere or outer sphere reac- the species involved as: tions, depending on the affinity of these ions for the surface (Hayes and Leckie, 1987). In general, cations KA = (aS.A) (an)" (21) having lower hydrolysis constants and higher charge (aA)(SsB)" " are more strongly adsorbed and undergo inner sphere Alternative formulations of this chemical reaction reactions to form coordination bonds with surface approach, such as the Gapon equation (Bolt, 1967), sites. Cations which adsorb to a weaker extent, such often employ concentrations instead of activities. as alkaline earth metals, undergo outer-sphere sur- Such formulations have been shown to agree with face complexation reactions which result in ion-pair experimental data over narrow concentration ranges. formation. Similarly, weak base anions form outer- The selectivity coefficients of the mass law approach sphere ion-pair bonds to surface sites and have the are related to ion size, ion charge and mineral type. lowest affinity for oxide surfaces, adsorbing at rela- Exchange affinities for major soil cations generally tively low pH values and only to the most acidic sites. increase with increasing charge and decreasing hy- drated radius, often follow the Irving-Williams series Vo vii Vd (Sposito, 1984a), and are sensitive to molecular % oB ad configuration. lop÷ Ca2+ NO3 Sorption reactions which occur on variable (vis-gz- NO~ pb~+ 01. !a 2+ vis fixed) charged surfaces, such as soil organic No; matter, mineral oxides (SiO2, A1203, TiO2 and o,'H; ~o; Ca2+ FeOOH) and on the edge sites of layered silicate Na+ minerals comprise another class of electrostatic inter- NO~ Na+ actions. The association of ions with these surfaces is O" Na÷ hypothesized to occur through surface complexation d Immobile I Diffuse or ligand exchange reactions analogous to those Layer Layer which occur in solution. The charges on these sur- Fig. 9. Schematic representation of an electrical triple layer faces arise most commonly through protonation and (adapted from Hayes and Leckie, 1987). Sorption phenomena 509

Conversely, strong base anions are postulated to 4.2.2. Hydrophobic sorption undergo ligand exchange reactions with surface Hydrophobic interactions comprise the primary hydroxyl groups. motivation for a large class of sorption reactions in Examples of pertinent binding relationships are the subsurface. The association of neutral, relatively presented in Table 2. As shown, a diprotic represen- nonpolar organic molecules with soils often results in tation of surface acidity has been assumed in which quasi-linear equilibrium sorption patterns, and the two inner sphere mass law equations can be written magnitude of the associated coefficients often vary to describe the amphoteric properties of the sur- with the organic carbon content of the soil. The face sites. In conjunction with these, a number of importance of soil organic matter in determining the outer-sphere electrolyte surface reactions have been extent of sorption of certain solutes, earlier evidenced proposed. by comparison of Figs 6 and 7, is dramatically The TLM model utilizes the Stern--Grahame model illustrated in Fig. 10, which shows the sorption of (Grahame, 1947) for description of the electrical lindane on a soil before and after the organic matter double layer and charge balances based on the reac- associated with that soil was removed with a strong tions presented in Table 2 for each of the planes. oxidant (Miller and Weber, 1986). Such observations Application of the model requires determination of suggest that the sorption reaction may well arise from the mass law constants, the surface density and "partitioning" of the solute into an organic phase on the inner and outer layer capacitances. Methods the surface of or within soil particles or aggregates. for determination of these parameters have been The linear isotherm distribution coefficients (KD) summarized elswhere (Kent et al., 1986). which result are often normalized by the fractional Successful applications of ion association models organic carbon content of the soil, ~o~, to give an for description of solution phase interactions have organic carbon normalized isotherm coefficient, Koj lent support to the notion that such an approach may provide an effective means for characterizing = (22) interactions between dissolved inorganic species and Ko~ KD sorbent surfaces. Reservations exist, however, particularly with respect to how well the model corre- The Ko~ then represents the hypothesized distribution sponds to actual mechanism in stoichiometric and coefficient for a sorbent composed entirely of organic energetic relationships (Westali and Hohl, 1980; carbon. Values for Ko~ have been found to vary by a Morel et al., 1981) and because of the inherent factor of only two or so for a wide range of soils and complexity of natural surfaces. sediments (Schwarzenbach and Westall, 1981). This

Table 2. Triple Layer Model Reactions and Equilibrium Expressions

Reaction Equilibrium Expression Surface protolysis reactions SOH~ ¢~, SOH + H + Kint_ ISOH] [H+] al- [SOH]; exp('F~0ART)

SOH ¢=) SO" + H + Kint _ [SO-][H+] a2- [SOH]; exp('l%U0/RT)

Electrolytesurface reactions SOH + Na + ¢=) SO--Na+ + H + Kint _ [SO-Na+][H +] Na - [SOH][Na+]; exp('F(~F0"~Pb)/RT)

int [SO'H; "NO3"] SOH +.+ + NO~SOH;-~O~ KNO 3 = [SOHI[H+ltNO3]~ cxp(F(V0"Vb)/RT)

Oumr-sphcre surfacereaction

Kint [SO-Ca2+][H+] SOH + Ca 2+ ¢~ SO--Ca 2+ + H + ~t- [SOHI[Ca2+]; exp('F(W0-2~Fb)/R'F)

Inner-sphere surface reaction

K~ [soPt~+}Iu+] SOH + Pb2+ ¢=* SOPb+ + H + Fo = +2 + ex~o~T) [SOH][Pb ]2 =

WR 25/5--B 510 WALTERJ. WEBERJ& et al.

correlation provides an approximation to the data trend, but that significant differences exist between "predicted" and observed values. ll20~'~ • Stripped More rigorous evaluations of the sorption of hydrophobic organic compounds by soils has related that sorption to the organic matter associated with the soils. The organic carbon normalized distribution coefficient has been related to the partitioning between the aqueous solvent phase and the organic phase [recall equation (7)]:

oc .... . (23) f,.2 Because of the relative uniformity of solute-soil 0 1000 2oo0 3o0o 4000 organic interactions compared to solute-aqueous Equilibrium SolutionConcentration (IJg/L) phase interactions, differences that exist between the Fig. 10. Sorption of lindane on Ann Arbor I soil before activity coefficients, f~. 1, of various solutes in aqueous (unstripped) and after (stripped) removal of soil organic phase are likely to be much more significant than matter in CMBR systems (after Miller and Weber, 1986). differences in the activity coefficients, f,:, for the same solutes in the sorbed phase (Karickhoff, 1984). variability has been linked to the nature of the Partitioning analyses of this type have been employed organic material in the soil (Garbarini and Lion, to relate the sorption of hydrophobic organic com- 1986) and to concurrent mineral-site sorption pounds to their partitioning behavior in octanol- (Karickhoff, 1984). The relative constancy of Ko¢ has water systems (Chiou et al., 1982) and their retention strengthened the notion that the sorption of hydro- patterns in reverse-phase liquid chromatography phobic organic compounds onto soils may be likened systems (Chin et al., 1988). to partitioning or absorption into a uniform organic The predictability of sorbed phase activity co- phase (Chiou et aL, 1983). efficients using Flory-Huggins polymer theory has The importance of solute hydrophobicity in also been examined. The organic phase is considered sorption reactions on soils and sediments has been as an ideal polymer and deviations from ideal be- confirmed qualitatively by numerous observations havior due to size disparities between solute and the that Ko~ values for particular solutes on a wide variety polymer molecules can be quantified (Chiou et al., of such natural sorbents can be correlated reasonably 1983). This calculation relates the sorbed phase well with the octanol/water partition coefficients of activity coefficient to the properties of the solute and those solutes. One such correlation developed by polymer and an interaction parameter. Schwarzenbach and Westall (1981) is compared in Predictions based on traditional partitioning Fig. 11 to our own experimental data for several theory frequently suffer to some degree from failure different solutes and soils. It is apparent that the to account for the presence and role of macromolecu-

i [] Michawye • Ann ArborI • Ann ArborII A A DeltaII [] I& • OSH Sediment --- Schwarzenbach& 2 Wcstall (1981 , I ) i 2 3 4 5 6 log Kow Fig. 11. Comparison for several different solutes of the relationship between partitioning coefficients for sorption by a variety of aquifer soils and sediments, normalized to organic carbon content, and the log octanol-water partition coet~cient. Sorption phenomena 511 lar dissolved organic matter in the solvent phase. 5. SORPTION RATE PROCESSES Natural "dissolved organic material" has been shown to increase the effective solubility of hydrophobic Equilibrium relationships comprise a set of limiting organic compounds (Carter and Suffet, 1983). This conditions for sorption processes, a set of conditions solubility enhancement has been ascribed to either predicated on there being sufficient time for a system alteration of the structure of the aqueous phase by the to achieve thermodynamic stability. In practical sys- organic material or to a partitioning of solute into tems, however, the time scales associated with attain- organic polymers (Chiou et al., 1986). This associ- ment of this condition may approximate or exceed ation can lead to a decrease in the extent of sorption time scales associated with changes in solute concen- of solutes on solid phases. Chin and Weber (1989) trations due to macroscopic transport processes (i.e. and Chin et al. (1991) have presented a three-phase advection and dispersion). Under such conditions, binding model formulated on the basis of a modified the rates at which equilibrium is approached may Flory-Huggins equation for a dispersed polymer significantly affect the process and the distribution of phase. Their comparisons of model predictions and contaminants among the phases of the system. experimental observations demonstrate, as might be Whereas the extent of sorption is dependent only on expected, that decreases in sorption attributable to the initial and final equilibirum states, rates of sorp- dispersed polymers in solution phase are most tion depend on the path leading from the initial to marked for relatively hydrophobic solutes. the final state. In porous media, these paths include The simple relationships discussed above for corre- events that are controlled either chemically or by lating hydrophobic solute sorption are applicable molecular-level mass transport. Molecular-level mass only to dilute aqueous systems. In severely contami- transfer refers in this context to stationary phase nated subsurface systems, the properties of the sol- diffusion processes, as differentiated from the fluid- vent itself may change. Such changes in aqueous associated macro-scale transport processes of advec- phase characteristics can be accounted for in the case tion (convection) and hydrodynamic dispersion. The of nonpolar molecules by relating the distribution influence of these latter processes is addressed in the coefficient in the mixed solvent system (KDM) to the co- final section of this paper. At this point, however, it efficient in the pure aqueous system (K w) through an is appropriate to consider in detail the two molecular expression developed from the general relationship processes that control rates of adsorption at the between the magnitude of the "hydrocarbonaceous" microscopic level; that is, reaction rate and local mass surface area of the solute (HSA) and its aqueous transfer. phase incompatibility, according to the so-called "solvophobic theory" (Nkedi-Kizza et al., 1985): 5.1. Reaction rate control K~ HSA/~c. (24) Chemical reaction rates depend on the nature of In K-~ = -~AT'-~ specific interaction(s) that occur between the solute The term ~ is an empirical coefficient, A7' is the and sorbent in the sorption process. Physical sorption difference between HSA/aqueous solvent, and processes are generally rapid, with local (site) equi- HSA/co-solvent interracial tensions, and ~c is the librium being achieved within milliseconds or, at fraction of the organic co-solvent in the aqueous most, seconds. The larger activation energies associ- phase. Equation (24) describes the log-linear relation- ated with other types of sorption (e.g. chemisorption) ship which has been observed between solute sorption may, however, lead to slower rates. The fundamental and the fraction of the co-solvent in the aqueous basis for molecular characterization of reaction phase. It is difficult to apply the equation directly kinetics is the law of mass action, which states that in many instances because the difference in these the rate of an elementary homogeneous chemical interracial tensions must be determined experimen- reaction is directly proportional to the product tally. Values for this quantity for several co-solvent of the masses (more rigorously, activities) of the systems have been presented (Nkedi-Kizza et al., reacting species. While sorption reactions are 1985). clearly not homogeneous and rarely elementary, it is Hydrophobic sorption models provide a con- possible to draw a stoichiometric analogy. The stoi- venient basis for prediction of sorption equilibria chiometry for a reaction involving a sorbate mol- only for those classes of compounds which meet the ecule, A, and a sorbent site, S, may be characterized assumed conditions. This implies relatively nonpolar, schematically as: neutral solutes, and sorbents similar to those with A + S~-~A-S. (25) which the correlations were developed; in general, soils and sediments with greater than 0.1% organic The term A-S represents the sorbed complex, de- carbon content. The sorptive behavior of polar and noted in this particular case as the association of one ionic organic solutes often manifest significant devi- mole of A with 1 mole of S. For the simple case in ations from the correlations presented, relating to which only one reacting species and only one type of differences in the forces responsible for the sorption site are involved in a one-step chemical reaction, the reactions. law of mass action analogy suggests that the forward 512 WALTER J. WEBER JR et al.

rate or "velocity" of sorption, VF, will be second The resulting expression for the overall rate of reac- order and of the form: tion, Vnct, is then:

VF = kF CA Cs. (26) dC^~s -- = v~t = k ~ C A - kRCA_ s. (34) dt The parameter kF is the time-independent constant of proportionality or rate constant, and CA and Cs For an element of saturated porous media having a represent the mass concentrations of A and S, porosity E and a solid phase density #s, and letting q respectively, at any time. Similarly, the rate of represent the quantity of sorbate, A, associated with desorption, vR, is defined in terms of a desorption a unit mass of sorbent, the rate of sorbate uptake can rate constant, kR, and the mass concentration be expressed as: of site-associated molecules of sorbent at any time, dq , E CA_S: dt = kF ps(1 - E----~CA - krtq. (35)

VR = kRCA_ s . (27) Rate expressions derived from mass action analogies The net rate of sorption, and thus the time rate of provide only first approximations to the true form of sorbate uptake in a completely mixed batch reactor sorption rate relationships. Experimentally deter- (CMBR) system, is then given by the difference be- mined reaction orders and coefficients seldom corre- tween the rates of the forward and reverse reactions, spond to those implied by the reaction stoichiometry or: and associated equilibrium energy state. The fact that experimental data for any particular system may be dCA s dCA fit by a given rate expression is not sufficient evidence l/he t ~ dt dt that the molecularity of the reaction is that implied by the rate expression. = VF--VR = kF CACs - kRCA s" (28) The simplicity of the rate expressions given above It is evident that rate models based on the laws results in part from the inherent assumption that of thermodynamics must have an equilibrium bound- chemically equivalent reaction rates obtain for all ary condition consistent with the assumptions under- sorption sites. The fact that natural sorbents fre- lying the rate law derivation. For the rate model quently exhibit functionally nonuniform surfaces expressed by equation (28), setting the left side to zero means that empirical application of the simplified rate (i.e. the net rate of sorption is zero at equilibrium) expressions frequently involves determination of yields: sample-averaged coefficients, which may not be applicable beyond the condition of experimental measure- k F CAC s = kp.CAs. (29) ment. To counter this potential deficiency, a number Upon rearrangement equation (29) gives an effective of heterogeneous reactive site rate models have been "equilibrium constant" for the stoichiometric anal- suggested. These take a variety of forms, some of ogy [equation (25)] drawn for this sorption reaction: which are summarized in equations (36)-(38) below. kF CA-s (1) Two site models, for example, where rapid kR CA Cs = K~q. (30) adsorption onto one type of site is ac- companied by a slower reaction onto a A mass balance on the total number of surface sites, second type of site: CT, gives: dq k~ l E Cr = CA_s + Cs. (31) d--t = p~(1 -- e~-"--)CA

When equations (30) and (31) are then combined, an E equation having the same general form as the Lang- +k~~CA-kRq. (36) muir isotherm results [compare to equation (11) for CA-S = qc, CT = Q0, and Keq = b]: (2) Models in which a continuous and specific range of site reactivities are hypothesized, C r K~q C A (32) such as the Elovich equation (Travis and CA-S = 1 + Keq C A" Eitner, 1981): A first-order approximation to the forward rate law given in equation (26) can be employed in cases = B 1 exp . (37) where the number of surface sites is sufficiently greater than the number of solute molecules that C s (3) Expressions similar to the simple rate laws in equation (26) can be considered constant. Thus, a with additional parameters included, such pseudo-first-order rate law is assumed for the forward as a nonlinear first-order rate equation: reaction rate: dq =k' E n -~ F ~ C - kRq. (38) V F = kFCsC A = k'FC A . (33) Sorption phenomena 513

These models usually derive from some presumed conceptual models described below (see, for mechanism and set of related assumptions. As em- example, various process model developments in phasized in the discussion of equilibrium isotherm Weber, 1972). models and reiterated above, however, the fact that they may give reasonable representation of 5.2.1. Conceptual models experimental data in any given application does Models for describing microscopic mass transfer not necessarily mean that the associated assump- are generally predicated on assumptions regarding tions and implied mechanisms are verified for the ~edominant or controlling transport mechanisms application. operating within specific types of media or domains. Microscopic mechanisms of mass transport in fluid 5.2. Mass transfer phases include diffusion of solute molecules through Effective rates of sorption in subsurface systems are elements of fluid and solute transport facilitated by frequently controlled by rates of solute transport molecular-scale movement of fluid elements at or rather than by sorption reactions per se. In general, within fluid phase interfaces (surface renewal) and mass transport and transfer processes operative in across microscopic velocity gradients (Taylor dis- subsurface environments may be categorized as either persion). The particular mass transfer mechanism "macroscopic" or "microscopic". In the content of which predominates in any situation depends on the this discussion, macroscopic transport refers to properties of the solute and the medium comprising movement of solute controlled by movement of bulk the domain, and on the microscopic hydrodynamics solvent, either by advection or hydrodynamic (mech- of the flow regime. Under fluid flow conditions anical) dispersion. By distinction, microscopic mass typical of subsurface systems, molecular diffusion transfer, the focus of the discussion, refers to move- generally dominates microscopic mass transfer. Mol- ment of solute under the influence of its own molecu- ecular diffusion can be either random ("Fickian") or lar or mass distribution. constrained ("Knudsen") by the boundaries of the One of the fundamental steps involved in charac- medium, such as surfaces bounding pore spaces. terizing and modeling microscopic mass transfer is Knudsen diffusion occurs when both molecular vel- appropriate representation of associated resistances ocities and ratios of longitudinal to radial pore or impedances, including relevant distances over lengths are high, and can be significant in gas phase which solute is transferred and relevant properties of mass transfer operations. Molecular diffusion in the medium through which transfer occurs. The liquid phase is, however, generally controlled by nature and characteristics of such resistances vary Fickian motion. In this type of diffusion, the velocity with local conditions associated with particular com- at which solute migrates along a linear path within a binations of sorbent, solute, fluid, and system particular coordinate system is directly proportional configuration. Differences in local conditions and to the gradient in its chemical potential, #i, along the associated transport phenomena are typified in sub- path; that is, to the thermodynamic "driving force". surface systems by differences between solute trans- As noted previously, the chemical potential of a port through the interstitial cracks and crevices of substance is directly related to its activity and, in rocks or soil particles, through organic polymer dilute aqueous solutions, to its concentration, Ci. It matrices associated with soils, and through internal follows then that the time rate of solute mass flow by fluid regions of soil aggregates. Mathematical de- diffusion along a path, x, and across a normal scriptions of microscopic impedances and mass (perpendicular) unit cross-sectional area (i.e. a one- transfer processes within fluid and sorbing phases are dimensional flux) is directly proportional to the so- generally structured upon one of several different lute velocity, and thus to the gradient concentration. types of conceptual models, tailored as necessary for For point-wise instantaneous diffusion, this flux, ~ ~, a particular circumstance by appropriate assump- is then given by: tions and constraints regarding initial and boundary ~C conditions and system behavior or state. F~,x = --Dl ":-. (39) These same conceptual models comprise the basis OX for a wide range of process descriptions in a variety Equation (39) expresses Fick's first law for dif- of natural and engineered environmental systems. fusion under non-steady state conditions. For Mass transfer typically controls overall rates of liquid phase diffusion the constant of propor- aeration and reaeration in natural waters; evolution tionality, D~, is termed the free liquid diffusion co- and dissolution of volatile compounds from solids efficient, most commonly referenced to the aqueous and liquids; dissolution of solid substances and non- phase. aqueous phase liquids into water; and treatment The driving force, ~C/~x, in equation (39) may operations involving stripping, solvent extraction, relate to either a constant or instantaneous difference membrane separations, ion exchange and adsorption in mass concentration across a homogeneous layer or processes. In such cases the specific modeling and/or "film" of fixed size (i.e. Ax - 6), or to a time-variable design relationships for these processes are pre- concentration profile along a continuous path of dicated on one or more of the several different variable length. The proportionality constant or 514 WAL'r~ J. Wrn~ JR et al. diffusion coefficient is affected by various factors difference between the mass fluxes into and out of which relate to molecular interactions between the that domain. Thus, for a domain of volume V, length solute and the solvent, including the size, configur- Ax, and unit cross-sectional area: ation and chemical structure of the diffusing molecule and the chemical structure and physical properties AC (e.g. viscosity) of the liquid. Values of D~ for diffusion of typical organic contaminants in pure aqueous In the limit (&~ ~ 0, At --. 0), equation (4t)), takes the solutions generally fall in the range 0.5-5.0 x form: 10-Scm2/s. It should be noted, however, that the diffusion of any solute through interfaciai aqueous (o_c =_o regions between fluid and/or sorbent phases may Ot lie c~x (-F~'x)" (41) involve resistances or impedances which differ from those of pure water. These differences arise because In Type I models the only impedance to mass transfer the properties of interfacial regions often reflect arises from the uniform resistance of a homogeneous molecular interactions between adjacent bulk phases. medium over a straight-line distance of travel. Adap- The magnitude of the diffusion coefficient for a tation of the general schematic given in Fig. 12 to this solute in an interracial domain reflects these vari- particular type of mass transfer is represented by ations. For example, the accumulation of molecules Fig. 13. In this case the flux associated with the other than those of the diffusing solute in an inter- microscopic mass transfer process is defined by Fick's face can increase resistance to transfer and yield a first law, equation (39), and equation (41) can be decreased diffusion coefficient, or drag forces near written: surfaces may effect reductions in diffusion coefficients in interfacial regions between liquid and solid phases. Models depicting microscopic molecular transport If there is no net accumulation of solute in the of conservative (non-reactive) substances in homo- diffusion domain (i.e. steady-state) the left-hand geneous or single-phase domains involve relatively side of equation (42) is zero and integration yields straightforward applications of equation (39). These a constant gradient in concentration across the are referred to here as Type I models. If the domain domain; that is, a linear driving force for mass is homogeneous but also involves a reaction of the transfer. If in a given system the spatial integration species being transported, the transport is described applies to a fixed distance Ax = 5, then a steady-state by a Type II model. Transport in heterogeneous constant flux over this distance can be represented domains (multi-phase) is described by either Type III in terms of the upgradient and downgradient bound- or Type IV models, depending upon whether solute ary concentrations, Co and C~ respectively, and a reactions are involved. solute velocity or mass transfer coefficient, kf, as follows: 5.2. I. 1. Type I domains and models -C0) Models to describe mass transfer rates in any (F~i,~)l~ffio = --Dl (C6 6 particular system typically incorporate the instantaneous point-form description of diffusion given in DI equation (39) in the appropriate mass balance or --~-(C0-c6)--kf(c 0-c6). (43) continuity equation for that system. For transport of a conservative solute through a diffusion domain It is apparent from equation (43) that this "linear such as that depicted schematically in Fig. 12, the driving force" model for mass transfer is similar in mass continuity relationship states that the time rate form to a first-order reaction rate equation, and its of change in mass within the domain is given by the solution can be approached in similar fashion.

Flux

I I o &x Distance, x Fig. 12. Schematic depiction of a diffusion domain for application of the mass continuity equation. Sorption phenomena 515

on chemical kinetics. For a first-order irreversible reaction with a rate constant, k, for example (aC/~t), = v = kC. As illustrated in Fig. 14 for steady-state conditions and boundary conditions identical to those given in Fig. 13 for the Type I model, the gradient in concentration, and thus the flux, varies across the domain as a result of the continued depletion of solute. For Impedance: steady-state transfer across a fixed diffusion domain, Solvent Resistance C o fi, coupled with a complete depletion of solute by first-order irreversible reaction within the domain, solution of the flux relationship at the upgradient C boundary of the domain yields: , C~ ,Co[ °' ] (F~, x)l x =0 = T Ltan-h ~)o.i.j (45) Ax: Distance, x Fig. 13. Schematic representation of solute migration and where 6(k/Dl)°5 in equation (45) is the Thiele modu- steady-state concentration profile for a Type I diffusion lus and the ratio [6(kDl)°'5/tanhcS(k/Dj) °'5] is the domain. dimensionless Hatta number, which characterizes the relative significance of the reaction and diffusion processes. As the Hatta number increases with in- It is imperative to note that the mass transfer creasing values of the reaction rate coefficient, k, the coefficient and concentration relationship given in denominator in equation (45) approaches unity and equation (43) have been developed for, and apply the flux at the boundary becomes: strictly only to, steady-state diffusion in a homogeneous medium in which there is no impedance to (F~i, x)lx ~ o = (kOl)°"Co. (46) diffusion other than that provided by resistance of the medium to movement of solute molecules. Other Comparison of equation (46) to equation (43) for a types of solute diffusion involving non-steady con- value of C~ = 0 indicates that the reaction term ditions, concurrent reactions, sorption and accumu- effectively renders the flux, and therefore the mass lation, and/or tortuous paths around obstacles, transfer coefficient, proportional to the square root of involve additional impedances and require different the free liquid diffusion coefficient rather than to model formulations. This is emphasized here because the diffusion coefficient itself. This underscores expressions of the same general form as that given in the level of error which might be introduced by equation (43) are often employed as expedients to using a Type I model formulation to represent a Type estimate mass transfer in more complex systems. II mass transfer process without specifically charac- When this is done, the gradient in concentration in terizing the relative effects of reaction(s) and mass the more complex system may in fact not remain transfer. constant, and the mass transfer coefficient may im- plicitly include factors other than just the free liquid diffusion coefficient and a fixed diffusion distance, factors not specifically identified and/or quantified. Under such circumstances the model, even if it can be fitted well to a particular data set, becomes a condition-specific relationship which may have limited EI,x ° utility for any application other than characterizing that data set.

5.2.1.2. Type H domains and models Impedances: In some cases of mass transfer a solute may be Solvent Resistance subject to depletion by reactions such as hydrolysis or Co Reaction(s) oxidation during the course of its diffusion through a particular domain. These reactions can be included in the continuity expression developed above, leading C~ to Type II models of the general form:

0i Distance, x A~ :5 Fig. 14. Schematic representation of solute migration and where (tiC~tit), is a reaction velocity rate relationship steady-state concentration profiles for a Type II diffusion of one of the types discussed previously in the section domain. 516 WALTERJ. WEnm~ JR et al.

5.2.1.3. Type III domains and models conditions is similar to that of Fig. 13 for diffusion Transport of a solute across a diffusion domain can in a Type I domain, and the flux relationship can be also be altered by constraining movement of the obtained in a manner analogous to that used to solute to specific flow paths. These conditions gener- develop equation (43), yielding: ally occur when diffusional transport through a het- (F~, x) Ix- 0 = ke (Co - C6). (49) erogeneous (multiple phase) medium is restricted to certain regions or is more rapid through particular The effective mass transfer coefficient, ke, in equation pathways, as illustrated schematically in Fig. 15. The (49) now combines the effective diffusivity of the mass transfer expression for diffusion in a Type III solute and the characteristic length of the domain. domain is similar to that for Type I, with the notable exception that the mass transfer coefficient now be- 5.2.1.4. Type IV domains and models comes a composite or "effective" parameter that The heterogeneous domains associated with Type reflects both the increased path length and the path IV mass transfer processes involve combinations of constrictions which alter transport of the solute in the the interactions described for Type II and Type III domain. The increased path length can be quantified models. As depicted in Fig. 16, the diffusion paths are as the ratio of the actual to the shortest path through tortuous and constricted and there are reactions such the domain, a ratio frequently termed the "tortuos- as sorption and catalyzed transformation with the ity" (Giddings, 1965). Constrictions of flow paths can surfaces of impermeable or semi-permeable phases in further increase impedance to transport. When solute the domain which further impede solute transport. transport is restricted to the fluid-filled fraction of the For the case of transport through a Type IV domain domain (volumetric water content), 0,, and both in which the only reaction is sorption of solute by a tortuosity and constriction effects are lumped in an solid phase of density Ps, the steady-state continuity apparent or effective diffusion coefficient, De, appli- equation can be expressed in a form analogous to cation of the mass continuity expression to the Type equation (44) for the Type II domain, but the reaction III domain leads to a modified form of equation (42): term (dC/dt), is now related to, and expressed in terms of, the rate of change of the solid phase (0.o concentration, q, as follows: (c3C) (1-,) aq If the void space is saturated with water, then the ~-,= , p~. (50) value of 0w is equal to the porosity, E, and equation (47) can be expressed as: The mass continuity expression for diffusion in a Type IV domain then combines equation (50) with the expression developed in equation (48) for the Type III domain, yielding: If the spatial gradient of porosity is very small, which (c3C) 1 0 (OC) (1-,)c3q is most commonly the case, E can be brought out of the derivative term and cancelled out in equation (48). The concentration profile shown in Fig. 15 for Again, e can be factored out of the first term on the diffusion to a Type III domain under steady-state right side of equation (53) if the spatial gradient of

F,°~ F I,x°

Impedances: Impedances: Solvent Resistance Solvent Resistance Co Co Tortuosity ~ . Reaction(s) C C % C~

Distance, x Ax~ 0 Distance, x A~ :8 Fig. 15. Schematic representation of solute migration and Fig. 16. Schematic representation of solute migration and steady-state concentration profile for a Type III diffusion steady-state concentration profile for a Type IV diffusion domain. domain. Sorption phenomena 517 porosity is very small. A general schematic concen- accumulation/depletion within, the adjacent domain. tration profile for steady-state diffusion in a Type IV This discussion focuses on situations involving ac- domain is shown in Fig. 16. It is apparent from cumulation in the downgradient domain because that equation (51) that the relative rates of solute sorption condition pertains most directly to microscopic trans- and diffusion dictate the extent to which the concen- port processes associated with sorption reactions in tration profile, and therefore flux, decreases with subsurface systems. distance through the domain pictured in Fig. 16. It is A simplified representation of diffusion through a also apparent that more strongly sorbed solutes Type I domain of fixed depth 6, cross-section A, and will exhibit a greater retardation in overall rates of constant upgradient boundary concentration into an migration through such domains. absorbing domain of volume V is shown in Fig. 17. The instantaneous rate of change in solute concen- 5.2.2. Application considerations tration, q, in an absorbing domain having a density Subsurface systems are often comprised by mul- Pv is given by: tiple diffusion domains of different types and degrees Oq A D 1 of impedance. As a consequence, mass transfer pro- .... (Co - C,O. (52) ~t Vpv 6 cesses associated with sorption reactions in such systems frequently involve two or more consecutive The accumulation of solute in the absorbing domain diffusion steps. It is generally the case, however, that results in a change in its concentration at the interface one of these steps is significantly slower than the between the two domains; that is, the downgradient others, and is therefore "rate determining" or "rate concentraion of the diffusion domain, C6. If the limiting". Identification and characterization of the sorption process is linear and a local equilibrium state step which controls overall rate in any given situation is maintained between the two domains, then the greatly facilitates the process of modeling. Indeed, instantaneous rate of change in Ca can be related to because of the potential mathematical and parameter 8q/Ot by a simple distribution coefficient, K o [see evaluation complexities otherwise involved, it is in equation (10) and ensuing discussion]. Thus: many cases an imperative aspect of model implemen- t3C6 ,4 D~ k. tation. Once a rate determining step has been - v 6Kop~ (Co- c,) = .-;=- (Co - c,). (53) ~t identified, models of the type presented above can ^nPv be adapted to the particular boundary and state The effective mass transfer coefficient, k:, in equation conditions appropriate for the system in question. (53) incorporates the cross-sectional area of the A further complication of real systems is that. diffusion domain, the diffusivity of the solute and the solute concentrations in domains of interest in such volume of the absorbing domain, and has the dimen- systems are generally time dependent, thus appli- sions of inverse time. If the sorption process is not cation of the steady-state forms of the various models developed above are seldom rigorously applicable. While departures from the boundary conditions associated with development of these models take various forms and yield different degrees of non- steady-state in real systems, practical approximations can frequently be made using certain quasi-steady- state approaches. 5.2.2.1. Quasi-steady-state models V The condition of true steady-state requires that the boundary concentrations of a domain, as well as the contributions of all sink and source terms, remain constant in time. If this requirement is not met, it may \ still be reasonable to assume a quasi-steady condition Co q(~ over time periods and/or for other specific conditions I for which the solute flux through the domain is large compared to the rate of change in boundary concen- c q trations. Quasi-steady-state modeling approaches are q(t1) predicated on the assumption that the concentration C~(t~) profile remains approximately linear throughout the domain, even though the concentration at one or both of the boundaries varies with time. The rate of 0 8 change of concentration at a domain boundary is t Distance, x usually dependent on the rate of concentration Fig. 17. Schematic representation of solute migration and change in the phase adjacent to the domain, which temporal concentration profiles for a Type I domain with may relate either to advective flow in/out of, or absorption in an adjacent region.

WR 25/5---C 518 WALTER J, WEBER JR et al. specifically characterized, experimentally measured values of k, may also incorporate by default the effect of solute partitioning. In any case, this coefficient, which generally must in fact be determined empirically, is highly system-specific and thus restricted in applicability. It is apparent from equation (53) that a number of Bulk concentration-independent factors impact the magni- Solution tude of (OCelOt). If these factors are such that the change in C~ over the period of interest is small relative to the difference in upgradient and downgradient concentrations (i.e. large KD, small k~), then solute migration through the diffusion domain can be approximated as a quasi-steady-state process. If this approach is applied to systems for which the assumption is inappropriate, the effective mass transfer coefficient k~ (or kJKt>) typically will be found to Cs vary with time, and thus be even further restricted in applicability to other than the exact circumstances for C Co S urface which it was measured. Models which incorporate Concentration such parameters are necessarily limited in their ability to predict mass transfer behavior for alternative C~ conditions, and are essentially restricted to data analysis and event simulation applications. 0 -'--I~ x r ,.,4.- 0 Quasi-steady-state mass transfer models based L upon assumptions similar to those discussed above Distance have been applied to describe the transfer of solute Fig. 18. Schematic representation and concentration profile between two different phases, the transport of solute for a film model to describe adsorption of solute by a between fluid regions through relatively small pores, spherical solid adsorbent. and solute transfer into intraparticle or aggregate regions. These models generally relate the flux through and out of one of the types of domains rate of change of the mass averaged (mass/mass) discussed to the rate of change in the solute concen- sorbent phase concentration, q, can be expressed as: tration of the adjacent phase, whether the phase is a dq_ 3 fluid or a solid. -- kf(C0 - C~). (54) Application of this modeling approach to descrip- dt Rpps tion of solute mass transfer across a hypothesized The driving force for mass transfer will obviously Type I immobile "boundary" layer of fluid immedi- vary as sorption proceeds, but if the gradient in ately adjacent to the external surfaces of a sorbent concentration across the film is steep and the sorbent yields the so-called "external film model". This par- strongly sorbing, the use of a quasi-steady-state ticular mass transfer step has been taken as rate model for describing flux across the film region is not determining in many modeling descriptions of solute unreasonable. uptake from bulk solution by liquid and solid sor- In this application the film transfer coefficient, kf, bents. As illustrated in Fig. 18 for adsorption from incorporates the diffusion of the solute in the medium solution at the exterior surfaces of a spherical solid surrounding the sorbent and the thickness of the film sorbent, "film" transport is governed by a combi- separating the sorbent surface from the bulk solution. nation of an effective boundary layer thickness and a When the boundary layer is a Type I domain com- molecular diffusivity of the solute within the layer. In prised by the same material (e.g. water) as the bulk practice, the concentrations, Co and C~, of solute at liquid, then the free liquid diffusivity can be em- the boundaries of the film in a modeling scenario ployed. The film thickness, however, is ill defined and similar to that depicted in Fig. 18 are typically taken not readily determinable. Indeed, in many ways "film as the solute concentration in the adjacent bulk fluid thickness" is more a conceptualization than a true phase and the solute concentration corresponding to physical dimension. The combined form, the mass instantaneous equilibrium with the solid phase at the transfer coefficient, kf, can in some cases be measured sorbent surface, respectively. The flux through the experimentally, or otherwise estimated by means of film is then expressed using a quasi-steady-state semi-empirical correlations which define functional model for a Type I domain of thickness 6 and cross- relationships between kf and such physical system sectional area controlled by the dimensions of the properties as mass flow rate, free liquid diffusivity of particle surface. If the spherical solid particle shown the sorbate, particle dimensions and in the case of in Fig. 18 has a radius Rp and density ps, then the time porous media, E, or porosity. Sorption phenomena 519

Quasi-steady-state models have been applied to phase density p, and a sorbed phase concentration q6 estimate solute flux into the internal voids of porous can then be written non-sorbing solids and other heterogeneous domains. koA, Goodknight et al. (1960) and Coats and Smith (1964), (Co - G) = ~(c0 - c~) for example, have employed such models to describe Y solute transfer into regions internal to an aggregated soil. In these instances, rates of change in concen- =E Ot ps(I-E) . (55) tration in the "immobile" internal fluid regions of the aggregates were described in terms of mass transfer The overall mass transfer coefficient, k,, incorporates through a Type III domain. A conceptualization of the effective cross-sectional area, A,, of the diffusion this type of model, sometimes termed a "bicontinuum domain; that is, the portion of the total cross-section model", is given in Fig, 19. of the domain available for fluid phase diffusion. It It is apparent that the representation given in also incorporates the volume, V~ = V, of the im- Fig. 19 assumes no preferential partitioning of solute mobile region, and thus has the dimensions of inverse between the mobile and immobile fluids, although time (similar in this regard to a first-order rate that condition can be accommodated by incorporat- coefficient). This coefficient is dependent on the physi- ing an appropriate partitioning relationship [(again, cal characteristics of the internal region (i.e. internal see equation (10) and related discussion]. Conceptual tortuosity) and the solute diffusivity, in accordance representations similar to that pictured in Fig. 19 with the Type III domain through which solute is have been applied to describe contaminant sorption transferred to the immobile region. and related transport in soil columns by assuming the A characteristic feature of the mobile/immobile overall rate limitation for removal of solute from region models discussed above is that they assume a bulk mobile fluid to be diffusion into immobile fluid linear concentration gradient across the diffusion regions associated with interstitial spaces or internal domain and a uniform internal concentration. In regions of soil aggregates (van Genuchten and reality, however, when the dimensions of the internal Wierenga, 1976). In such cases, the flux into the region and the flow path are of similar magnitudes, immobile regions under quasi-steady-state conditions for example, it is unlikely that the dual assumptions is equal to the rate of change in the sum of the of steady-state transport within the diffusion domain immobile fluid and sorbed phase concentrations. and uniform concentration within the internal region If the contaminant is distributed uniformly through- will be met (Coats and Smith, 1964). In such in- out the immobile region, the fluid phase con- stances, the overall mass transfer coefficients associ- centration of the region is characterized by the ated with models which do not specifically concentration, Co, at its interface with the diffusion accommodate these conditions will vary with time as domain. The solute accumulation in an immobile continued transport into the immobile region con- solute sorbing region having a porosity E, a solid- tinues (Rao et aL, 1980a). The same limitations apply to models structured on the basis of transport in Type IV domains. Application of quasi-steady-state assumptions are in these instances further constrained to situations in which either the reaction rates or thermodynamics yield only small changes in solute concentrations in the immobile region relative to the flux across the domain. As an approximation for either Type III or Type IV domains, it may be reasonable to consider the flux into the heterogeneous domain within regions of a particle or aggregate proportional to the difference between the external or mobile concentration and a volume averaged internal or immobile concentration (Gluekhauf and Coates, 1947). While this approximation may be valid for certain situations, sorption and/or accumulation must be taken into account. As a result, applications of quasi-steady-state models in such situations again C [ C~ often yield highly system-specific approximations.

5.2.2.2. Non-steady-state models More accurate and general representations of com- 0 5 i I plex mass transfer processes may be obtained by Distance, x employing non-steady-state models to describe solute Fig. 19. Schematic representation and concentration profile transport. The continuity expression for mass for a "mobile/immobile" domain or bicontinuum model. transfer by diffusion within the domain leads to 520 WALTERJ. WEBERJR et al. expressions similar to equation (44). Analytical sol- In application of the non-steady-state represen- utions to such equations under non-steady-state con- tation to a Type II domain, which is characterized by ditions are available for certain applications and a homogeneous diffusion medium and a solute reac- boundary conditions (Crank, 1975). In many cases, tion, the flux will represent a superposition of the however, numerical solutions are required. effects of diffusion and reaction. Solution of equation Equation (44) is in fact the continuity expression (46) for the non-steady-state condition for a first- which obtains upon application of the non-steady- order reaction with rate constant, k, and for hound- state diffusion equation to a homogeneous Type I ary conditions equivalent to those used to obtain domain. For a constant concentration boundary equation (56) (i.e. a constant concentration, Co, at condition, C = Co at x = 0, and a negligible concen- x = 0) can be expressed (Crank, 1975): tration at a large distance into the mass transfer region (a condition which may be met at the early stages of mass transfer), the concentration at a distance x into the region at any time, t, after initiation of diffusion is (Crank, 1975):

C = CO I1 - erf(~)]. (56) Under several limiting conditions, the flux into a Type II domain reduces to one of the solutions presented The concentration profile for this solution to the earlier. When the solute reacts very rapidly after it non-steady-state condition, depicted in Fig. 20, is enters the domain, the corresponding solution for the non-linear. The concentration profile reflects a mass flux at the boundary can be closely estimated by: flux which varies temporally and spatially within the F~i I x ffi o = (kDt)°"5 Co. (59) phase. It can be shown by substitution of equation (56) into Fick's law [equation (39)] that shortly This is identical to the steady-state solution for the after initiation of diffusion the flux of solute at the Type II domain presented in equation (46). When the upgradient boundary of the domain (x = 0) is: product kt is very small, the flux at the boundary can be expressed as: (57) F~ Ix= 0 = (1 + kt) C o. (60) In contrast to the flux relationship for steady-state diffusion in a Type I domain [equation (43)], the Equation (60) reduces to the solution for diffusion in effective mass transfer coefficient, k e, for non-steady- a Type I domain without reaction [equation (57)] as state diffusion is proportional to the square root of k approaches zero. the free liquid diffusion coefficient. The time depen- Solute transport in subsurface systems often in- dence of the mass transfer coefficient reflects the cludes non-steady-state movement into particles or decrease in the local concentration gradient as the aggregates. A heterogeneous Type III domain rep- solute accumulates, a condition which was presented resentation, with an effective diffusion coefficient earlier as a possible shortcoming of the steady-state which accommodates the tortuous flow paths which model. occur within such a domain, can be applied to describe this transport. In these instances, the geometry of the domain must also be incorporated into the non-steady-state model because accumulation within the domain, and subsequent decreases in the local concentration gradient, will be affected by E ° changes in fluid volume along the mass transfer path. Particles and aggregates in subsurface systems are commonly idealized as spheres in sorption rate models. In spherical coordinates, the non-steady- state continuity expression for non-reactive diffusion in a Type III domain of porosity ~ has the form: Co

=~-~r ~,r De-~r ). (61)

As indicated above, the effective diffusion coefficient, De, in equation (61) reflects the properties of the medium through which the transfer is occurring. Solutions to equation (61) have been presented by 0 Ax=8 I Distance, x i Crank (1975) for several different conditions. While Fig. 20. General representation of concentration profiles for Type III domain models such as that given in non-steady-state models. equation (61) have been applied for description of the Sorption phenomena 521

non-steady-state transport of non-reactive solutes in within the particle or aggregate. Such "dual" re- soil systems (Rao et al., 1980b), sorbing contaminants sistance models have been employed successfully to require application of Type IV domain models. As describe sorption of organic contaminants on soils at discussed earlier, fluxes across Type IV domains both the particle level (Miller and Weber, 1986) and reflect solute accumulation and retardation due to the aggregate level (Hutzler et ai., 1984; Roberts et aL, sorption by the solid phase, and expressions to 1987). A schematic representation of the series do- describe non-steady-state diffusion in such domains main concept and the concentration profiles associ- must account for the solute in both sorbed and ated with this type of model are shown in Fig. 21. The solution phases throughout the geometry of interest. application of a dual resistance model can incorpor- The non-steady-state continuity expression for ate both equation (43) for an external or film transfer diffusion and adsorption in a Type IV domain is, in and equation (61) to describe the internal transfer. spherical coordinates: The flux across the external film must be equated to the internal flux at the particle or aggregate surface. ~-[ v='~E~r 'r2D~ - E ,o.~-7. (62) Combining equation (43) with the general expression for internal flux leads to the condition: As noted in earlier discussions of steady-state models dq, for Type III and IV domains, E can usually be kf(C0 - C6) = DoPs ~r at r = Rp. (63) brought out of the spatial derivative and cancelled out of that term in equations (61) and (62). This condition must be met at the exterior of the Movement of solute in Type IV domains can occur particle or aggregate, and thus comprises a boundary both through diffusion of solute in liquid-filled re- condition for interfacing the solid and liquid material gions and migration of sorbate along sorbent sur- balance equations. faces. When sorbed phase concentrations are It is evident from the foregoing that various adap- sufficiently high, transport of solute by the latter tations of one or more of the basic "domain" models route can be a significant, or even dominant, part of discussed at the outset of our consideration of micro- the overall intraparticle or intraaggregate solute flux. scopic mass transfer have been employed to charac- Because pore and sorbed phase diffusion act in terize rate-limited sorption processes in different parallel, that process which results in the greater flux theoretical and experimental investigations. The is generally rate determining. Pore and sorbed phase choice of any particular model, the most "appropri- (surface) diffusion models have been used singly and ate" model, is generally predicated on the level of in combination to describe organic sorption rates for detail available regarding a given application, and the microporous sorbents, individual soil particles and level of "correctness" with which sorption processes soil aggregates (Weber and Crittenden, 1975; Weber, must be characterized and evaluated. These same 1984; Crittenden et al., 1986; Roberts et al., 1987; considerations of course govern whether any micro- Miller and Weber, 1988). Other descriptions of simul- scopic mass transfer model is to be preferred over a taueous diffusion and sorption in microporous sor- more simple "reaction rate" model; indeed, whether bents and soil systems have been made using it is even necessary or appropriate to account for the generalized expressions to represent both pore and time dependence of sorption processes in a particular sorbed phase diffusion (Weber and Ruiner, 1965; Wu application. Such issues go beyond considerations of and Gschwend, 1986). Attempting to distinguish microscopic mass transfer. They are crucial issues rigorously between sorbed phase and pore liquid intraparticle or intraaggregate diffusion may in fact be problematic for many combinations of soils and solutes, in part because neither the pore nor sorbed phase diffusion coefficient has a significant concentration dependence when the sorption isotherms are only weakly non-linear (Crittenden et al., 1986). Thus in many cases a microporous particle is treated as a homogeneous entity, the q expressed as e I Domain L Type IV Domain a bulk particle value, and the effective diffusion -2Co coefficient, De, treated as a bulk particle property. In many cases the overall mass transfer of solute c into and through Type II and Type IV domains can be controlled by a combination of resistances, and more accurate descriptions of sorption rates are therefore obtained by combining two (or more) mass transfer

models. A common conceptualization of sorption 8 ~.~- R processes involving microporous soil particles and ,0 ~ x Distance r -.,c--- 0 aggregates combines an external model for film trans- Fig. 21. Schematic representation and concentration profile fer with an internal model for diffusion/sorption for a dual resistance model. 522 WALTERJ. WEBERJR et al. with respect to the ultimate goal of characterizing and The term D h in equation (64) is a second-rank hydro- quantifying contaminant behavior in subsurface en- dynamic dispersion tensor, v is a pore-velocity vector, vironments, however, and must be considered in that C is the solution-phase concentration of solute, and context. To that end, the final section of the paper S(C) is a fluid-phase solute source term. When addresses the potential effects of sorption processes microscale processes are significant, equation (64) is on contaminant fate and transport in several typical expanded to include descriptions of reactions which circumstances. affect solute concentration, yielding the so-called advection--dispersion-reaction (ADR) equation: 6. SORPTION PROCESSES IN A FATE AND aC TRANSPORT PERSPECTIVE -- = -v. grad C + div(D h" grad C) at The sorption rate and equilibrium models presented above, whether mechanistic or phenomeno- + ~ +S(C). (65) logical, have been developed on what may be termed r a local or microscopic scale; that is, by describing The right-hand term subscripted with an r in equation processes at a molecular or particle level. Their (65) denotes the time-rate of change in concentration ultimate utility for characterizing and predicting the associated with a microscale reaction process, such as behavior and eventual fate of contaminants in sub- adsorption, which may in turn be represented by surface systems depends on: (1) the relative import- reaction rate models or by models describing micro- ance of sorption processes in the context of other scale mass transfer processes. The significance of reaction and transport processes operative in subsur- any particular microscale reaction on macroscopic face environments; and (2) our ability to determine solute transport can be estimated by: (1) conducting the level of complexity required to describe accurately controlled investigations to determine rate and the impact of microscopic processes on overall fate equilibrium parameters for that reaction; and (2) and transport at the macroscopic scale. It is not incorporating these parameters into the reaction term within the scope of this paper to examine macro- in the ADR equation and performing field-scale scopic transport models in detail, but there is value sensitivity analyses to determine the relative impact in considering different levels of model sophistication of the reaction, vis-d-vis the advection and dispersion required to capture the effects of sorption processes processes. and reflect them in predictions or estimations of In laboratory and controlled field-scale investi- solute transport under field-scale conditions. To do gations of transport and transformation processes this, several examples are selected to demonstrate the experimental design is commonly structured in a that adequate macroscale characterization of solute manner that will allow close approximation of system behavior in typical subsurface environments requires behavior with a one-dimensional form of equation thoughtful consideration and choice of appropriate (65). There are certain field applications in which use microscale sorption models. of a one-dimensional form of the ADR equation may Macroscopic models for tranport in subsurface also be justified, although this simplification generally systems incorporate advection and dispersion pro- does not provide sufficiently accurate representations cesses, and are generally structured on principles of of field-scale transport. Nonetheless, the one-dimen- mass conservation applied on a "volume average" or sional simplification does afford a convenient means otherwise statistically averaged basis. Generated on for evaluating the relative effects of reaction and a differential scale, the continuity relationship yields macrotransport processes in various contamination the following advection-dispersion equation: scenarios. Consider, for example, the finite control dC volume, V, represented schematically in Fig. 22 for a a--t- = -v.grad C + div(Dh.grad C) + S(C). (64) system of fluid flow through a porous matrix com-

L • I

Bulk Flow In ~ Bulk Flow Out Flux In, F? v z Flux 1,z z--O Out, F ? I i,z z=L Oisp ion t

"Toss or Generation by Reaction and/or Sorption Fig. 22. Control volume for one-dimensional transport through porous media. Sorption phenomena 523 prised by a stationary sorbent phase. The flux of Table 3. Forms of the one-dimensional advection-dispersion equation incorporating sorption equilibria and rate expressions dissolved fluid-phase component, i, entering or leav- General equation ing the control volume includes advective and disper- sive components. Within the control volume, the (0 component or solute of interest may undergo reaction (transformation) and/or sorption (phase transfer/ Local equilibrium exchange) with the sorbent phase. If equation (SO) is (ii) used to describe the sorption reaction term, and no other fluid-phase reaction or source terms are con- with linear isotherm sidered, the ADR relationship given in equation (64) (iii) simplifies in one-dimensional (z) form to an advec- tiondispersion-sorption (ADS) model: with Freundlich isotherm (1-c) 7K~nCn+ (iv) (66) First -order sorptionldesorption

The hydrodynamic dispersion term, D,,, in equation (1 -Qk q ~=q~-“~-k,C+&-- R 69 (66) is now a simple numerical coefficient, v, is the t component of fluid-phase pore velocity in the z Equilibrium and first -order sorption/desorption direction and qi is the volume averaged sorbed-phase ac solute concentration of component i expressed as a d,'Dh$-"g-ps !i$g~-I(Fc+ps (y k,q (vi) mass ratio. It follows from earlier sections of this paper that Mobile/immobile with quasi-steady -state linear driving force and sorption the sorption term, aqi/at, can assume a variety of forms comprising different rate and equilibrium components. Several examples of one-dimensional macroscopic transport models incorporating various - pI y B,% - % k,(C, - C,) (vii) equilibrium and microscopic rate models are summarized in Table 3. The most simplistic approach to where representation of sorption phenomena in a contami- k(C -C)=p(l-c)%+,s oM d I at at nant transport model is to assume that the time scales associated with the microscopic processes of diffusion and and sorption are very much smaller than those associ- (eh.4 = (C)w = 6 ated with the macroscopic processes of fluid trans- (&)M = (PA4 = PS port. This effectively assumes that equilibrium Internal diffusion prevails locally; that is, that the time rate of change of the sorbed phase concentration, qi, at any point z (viii) is instantaneously reflected in the time rate of change where of the solution phase concentration, Ci, at that point. This yields the so-called local equilibrium model q,r’dr

(LEM). A further simplification is to assume that the and relationship between q1 and Ci involves a direct proportionality of the type typically associated with simple partitioning or absorption processes. Express- ing this proportionality in terms of the distribution Dual resistance coefficient, K,, defined by equation (10) gives the --kk,(C-C,) (ix) following relationship: and

!$=L’ (67) at ?a? where The macroscopic transport model which results upon rearrangement of equation (66) and substi- :(C-C,)=D,$$ at r=R, tution of the relationship for (aqjat), given in and equation (67) is termed the linear local equilibrium c = Cl,,, in Fig 21 model (LLEM):

P&l - 0 1+ --Ku)(;):=R&$ 6 Equation (68) provides a reasonable means for first- cut assessment of the potential impact of sorption processes under field-scale conditions. For example, Fig. 23 presents an LLEM field-scale simulation 524 WALTERJ. WEe~ JR et al.

500 tation of the effects of sorption processes on contami- ConservativeSolute nant transport. Inclusion of more sophisticated NO Dispc~ol non-linear equilibrium models often provides better 400- Diction representation of sorption phenomena. Consider, for example, the potential error associated with macro- 300 scopic transport model predictions employing linear partitioning models in a system for which the actual equilibrium data are better characterized by a non- "~ 200. Non-ConservativeSolute linear isotherm model. For this particular example we ..... No Dispersion will assume that local equilibrium conditions prevail; 100. Dispersion this to examine singly the effects which accrue to the choice of the equilibrium model itself. The solute-soil system selected for analysis is comprised of tetra- 500 1000 1500 2000 chloroethylene (TTCE) and Wagner soil, for which Time (hour) CMBR equilibrium data obtained in our laboratories Fig. 23. Ten-meter field-scale simulations for a moderately are presented in Fig. 24(a). TTCE, a slightly polar hydrophobic contaminant using simplified forms of the chlorinated solvent of high volatility, has a moderate ADS transport model. degree of hydrophobicity (log Kow = 2.8) and the Wagner soil an organic carbon content of 1.2%. of the concentration profiles for subsurface trans- Figure 24(a) shows the "best fits" to the equilibrium port of a moderately hydrophobic solute through a data afforded by linear regression with a simple moderately low organic content soil under represen- partitioning model and by non-linear regression with tative conditions of fluid flow and hydrodynamic the Freundlich model. Clearly the Freundlich model dispersion. The fluid velocity and dispersivity are provides a better overall representation of the data, assumed constant thoughout the domain to further although portions of that data are reasonably well fit simplify the evaluation. These profiles represent con- by the linear model. Projection of TTCE transport in centration patterns at a point 10-m downgradient of a 1-m field-scale simulation made using two different a pulse input of contaminant as a function of time forms of the ADS transport model incorporating after addition of that input. Parmeter values utilized these two alternative isotherm models [see equations in this 10-m simulation are tabulated in Table 4. (iii) and (iv) in Table 3] calibrated with the data given Concentration-time profiles simulated by neglecting in Table 4 are presented in Fig. 24(b). It is apparent sorption and/or dispersion with all other conditions from comparison of these simulations that the use of the same are also presented in Fig. 23. Comparison a linear relationship to represent the equilibrium of these several profiles demonstrates that both sorp- sorption behavior of the TTCE with respect to the tion and dispersion must be accounted for in the Wagner soil results in a substantially different projec- one-dimensional ADS equation to adequately de- tion for contaminant transport than does the use of scribe contaminant distribution in systems where the Freundlich isotherm model. The most significant these processes are operative at the levels represented difference between the two projections is the slower in this simulation, which are reasonably typical of rate of travel of the center of mass when the Fre- field-scale circumstances. undlich model is employed. This is reflective of the Although the LLEM version of the ADS equation model's ability to account for the higher sorption has been widely employed for describing solute re- capacities observed at lower solution-phase concen- tardation by sorption in subsurface systems (Faust trations [Fig. 24(a)]. The increased retention at low and Mercer, 1980; McCarty et al., 1981; Pinder, concentrations and the decreased slope of the Fre- 1984), it has become increasingly apparent that this undlich model fit leads to greater asymmetry or model frequently fails to provide adequate represen- "tailing" of the solute pulse. It is important to note

Table 4. Field Scale Case Model Input Parameters

Parameter Value Units

Inital Concentration 1000

Velo~ty 1 meter/day

Dispersion Coefficient 0.1 meter2/day

Time of Solute Input 1 day

Soft Density 2.67 gr/cm3

Void Fraction 0.4 Sorption phenomena 525

100 to take account of these additive effects. The relative contributions of the reaction rate and hydrodynamic dispersion mechanisms are dependent on flow velocity and microporous particle or aggregate radius. The sensitivity of this apparent dispersion coefficient to particle or aggregate size and the related effective ~=- internal diffusion coefficient, De, for a Type IV domain is shown in Fig. 25, which was generated from a mathematical relationship between internal diffusion and hydrodynamic dispersion developed by ..'°°"* ..... LINEAR Parker and Valocchi (1968). This relationship demon- FREUNDLICH strates that dispersion due to mass transfer will dominate for systems comprised of large micro- .1 ...... i ...... i ...... 10 1 O0 1000 10000 porous particles or aggregates. In a similar manner, Ce (pg/I) it can be shown that mass transfer dominates at a. Sorptionisotherms higher flow velocities. Sensitivity analyses performed by a number of investigators have shown that use of 25 an apparent dispersion coefficient reasonably repro- duces contaminant breakthrough profiles for large 2o /; Peclet numbers. In contrast, this approach fails to describe accurately the asymmetry due to mass trans- ...- fer effects for systems dominated by hydrodynamic dispersion. .~ [ ...... LINF.AR FREUNDLICH The effect of rates on concentration profile asymmetry is apparent for chemically controlled rate phenomena as well as for mass transfer controlled sorption. These effects are illustrated in Fig. 26(a). 5- Equilibrium conditions for this particular example were described using a linear isotherm model, and a

0 " range of first-order rate constants was tested. The 50 100 150 200 250 300 ADS equations corresponding to the linear local Time (days) equilibrium and first-order rate representations for b, Resulting fi¢|d scale effects this system are equations (iii) and (iv), respectively, in Fig. 24. Sorption isotherms (a) and l-m field-scale simulations (b) for TTCE and Wagner soil using these isotherms Table 3 (Miller and Weber, 1988). These models, in the ADS transport model. calibrated for the conditions described in Table 4, yield the l-m field-scale simulations presented in Fig. 26(a). Comparison of the simulations clearly that these effects are significant even though the data demonstrates the potential error that might be associ- are not grossly ill-fit by the linear model. Indeed, the ated with the use of a local equilibrium model for a linear model would seem quite adequate had the system in which sorption rates are not truly negligible experimental data set been limited to a narrow range, say to only those data above about 150/zg/1. The importance of both measuring and accurately repre- 10 6 Microporous Particle or senting equilibrium sorption data over the entire 105 ~ AggmgamRadius range of interest in any given situation should thus be 104 ~ 10 mm abundantly clear. 10 3 ***** ----- I mm { _ _ . 0.,om The second consideration to be made here relates 10 2 '~, ,~~ to the validity of assuming that sorption time scales I01 "~,~ are not important with respect to transport time scales. A number of investigations have shown that field-scale contaminant transport may be rate con- 10 .2 -,~ ~ trolled (for a review see Brusseau and Rao, 1989). In v= 1 mid .,., --. ~ such cases, overall contaminant dispersion is due to 10-3 Rf = 5 ,.. 0[M = 0,5 "~" a combination of macroscopic and microscopic 10-4 ,~ effects. The most simplistic approach for incorporat- I0 -5 ing rate phenomena into model descriptions of con- ..... ;'o ..... 1: ...... ;'o ...... 1: ..... taminant transport is to assume that macroscopic and De (cm2/s) microscopic effects on front spreading are additive, Fig. 25. Relationship between effective intraparticle molecu- and that an effective or apparent dispersion co- lar diffusion (D,) and hydrodynamic dispersion (De) to give efficient, Dh.a, can be incorported into equation (68) equivalent effects on initial spreading. 526 WALTER J. WEBER JR et al. at the time scale of associated transport processes. It rate-controlled sorption process in this case translates is readily apparent that rate limitations for the sorp- into more than a 100% increase in the volume of tion process result in earlier arrival of the contami- contaminated water which must be treated. nant front at the 1-m downgradient point, as well as increased tailing of the front to increase the amount 6.1. Closure of total "dispersion" of the contaminant plume com- Sorption processes in subsurface systems are compared to the LLEM model, which accounts only for plex, often involving non-linear phase relationships macroscopic or hydrodynamic dispersion. and rate-limited conditions. We have demonstrated The conditions employed in developing the simu- that these processes impact reactive solute behavior lations for Fig. 26(a) represent reasonable values for under typical field-scale conditions, and must there- assessing the anticipated effects of rate-controlled fore be considered in attempts to model or otherwise sorption under typical background flow conditions. predict contaminant fate and transport in the subsur- Such non-equilibrium effects may play an even more face. It is further apparent from the examples we have important role in flow situations associated with considered that the thoughtful selection of appropri- typical "pump and treat" remediation strategies. ate microscopic equilibrium and rate models, models Consider, for example, the impact of rate-limited which adequately describe the inherently complex sorption on the volume of water to be treated in a and system-specific dynamics of sorption processes, remediation scheme in which the fluid velocity is is an imperative for accurate fate and transport increased by pumping to 10 m/d. As shown in Fig. modeling. 26(b), the effect of non-equilibrium conditions at a particular rate constant are even more significant in Acknowledgements--The general subject matter covered in this scenario, and the potential error associated with this paper was the topic of an invited lecture by the senior erroneously employing a LLEM even greater. The author (W.J.W. Jr) at an Advanced Studies Institute sponsored by the North Atlantic Treaty Organization at Washington State University in July 1989. The other 30" authors (L.E.K. and P.M.M.) were instrumental in helping EQUmmRRn~ to prepare visual and textual material for that lecture. A ..... k=O.~:~ Subsequently (October 1989) the senior author was asked to Ir,,I ...... k - 0.05/In" present the 1990 Distinguished Lecture Series sponsored by li..~l ...... k-o.ol~ the Association of Environmental Engineering Professors, 20 !" %1'JI ~=t~W and chose to expand upon his NATO lecture, again enlisting the assistance of Lynn E. Katz and Paul M. McGinley. This 4 paper is the combined result of the NATO and AEEP lectures.

REFERENCES

Barrow N. J. and Bowden J. W. (1987) A comparison of models for describing the adsorption of anions on a variable charge mineral surface. J. Colloid Interface Sci. 0 2000 4000 6000 119, 236-250. Time (hours) Bolt G. H. (1967) Cation-exchange equations used in soil a. Background flow conditions sciencela review. Neth. J. Agric. Sci. 15, 81-103. Brusseau M. L. and Rao P. S. C. (1989) Sorption non- 300 ideality during organic contaminant transport in porous EQUR.mRIUM media. CRC Crit. Rev. envir. Control 19, 22-99. A ..... k = 0.1/ht Carter C. W. and Suffet I. H. (1983) Interactions between ...... k = 0.05/hr dissolved humic and fulvic acids and pollutants in aquatic I I ~=10~Vd environments. In Fate of Chemicals in the Environment. American Chemical Society, Washington, D.C. 200 I~~1 ----D= 1 m2/d Chin Y. P. and Weber W. J. Jr (1989) Estimating the effects of dispersed organic polymers on the sorption of contaminants by natural solids I: a predictive thermodynamic humic substance-organic solute interaction model. Envir. Sci. Technol. 23, 978-984. Chin Y. P., Peven C. S. and Weber W. J. Jr (1988) Estimating soil/sediment partition coefficients for organic /i compounds by high performance reverse phase liquid fl I P~ chromatography. Wat. Res. 22, 873-881. Chin Y. P., Weber W. J. Jr and Eadie B. J. (1990) Estimating the effects of dispersed organic polymers on the sorption 0 ...... of contaminants by natural solids II: sorption in the 200 400 600 Time (hours) presence of humic and other natural macromolecules. Envir. Sci. Technol. 24, 837-842. b. Pump and treat remediation conditions Chiou C. T., Porter P. E. and Schmedding D. W. (1983) Fig. 26. One-meter field-scale simulations showing effects of Partitioning equilibria of nonionic compounds between tint-order reaction rates in typical flow (a) and remediation soil organic matter and water. Envir. Sci. Technol. 17, flow (b) situations. 227-231. Sorption phenomena 527

Chiou C. T., Schmedding D. W. and Manes M. (1982) Kent D. B. Tripathi V. S., Ball N. B. and Leekie J. O. (1986) Partitioning of organic compounds in octanol-water Surface-complexation modeling of radionuclide partition- systems. Envir. Sci. Technol. 16, 4-10. ing in grotmdwaters. Progress Report, Contract No. Chiou C. T., Malcolm R. L., Brinton T. I. and Kile D. E. SNL-25-1891, Sandia National Laboratory. (1986) Water solubility enhancement of some organic Kiselev A. V. (1985) Non-specific and specific interactions pollutants and pesticides by dissolved humic and fulvic of different electronic structures with solid surfaces. Disc. acids. Ent~ir. Sci. Technol. 20, 1231-1234. Faraday Soc. 40, 205-231. Coats K. H. and Smith B. D. (1964) Dead-end pore volume McCarty P. L., Reinhard M. and Rittmann B. E. (1981) and dispersion in porous media. Soc. Petrol. Engrs J. 4, Trace organics in ground water. Envir. Sci. Technol. 15, 73-84. 40-51. Crank J. (1975) The Mathematics of Diffusion. Clarendon Miller C. T. and Weber W. J. Jr (1984) Modeling organic Press, Oxford. contaminant partitioning in ground water systems. Crittenden J. C., Hutzler N. J., Geyer D. G., Oravitz J. L. Ground Wat. 22, 584-592. and Friedman G. (1986) Transport of organic compounds Miller C. T. and Weber W. J. Jr (1986) Sorption of with saturated groundwater flow: model development and hydrophobic organic pollutants in saturated soil systems. parameter sensitivity. Wat. Resour. Res. 22, 271-284. J. contain. Hydrol. 1, 243-261. Davis J. A. and Leckie J. O. (1978) Surface ionization Miller C. T. and Weber W. J. Jr (1988) Modeling and complexation at the oxide/water interface. II. Sur- the sorption of hydrophobic contaminants by aquifer face properties of amorphous iron oxyhydroxide and materials--II. Soil--column reactor systems. Wat. Res. 22, adsorption of metal ions. J. Colloid Interface Sci. 67, 465-474. 90-107. Morel F. M. M., Yeasted J. G. and Westall J. (1981) Eriksson E. (1952) Cation exchange equilibria on clay Adsorption models: a mathematical analysis in the frame- minerals. Soil Sci. 74, 103-I 13. work of general equilibrium calculations. In Adsorption of Faust C. R. and Mercer J. W. (1980) Ground-water mod- Inorganics at Solid-Liquid Interfaces (Edited by Anderson elling: recent developments. Ground Wat. 18, 569-577. M. A. and Rubin A. J.). Ann Arbor Science, Ann Arbor, Garbarini D. R. and Lion L. W. (1986) Influence of the Mich. nature of soil organics on the sorption of toluene and Nemethy G. and Scheraga H. A. (1962) Structure of water trichloroethylene. Envir. Sei. Technol. 20, 1263-1269. and hydrophobic bonding in proteins. II: model for the Gasser R. P. H. (1985) An Introduction to Chemisorption and thermodynamic properties of aqueous solutions of hydro- Catalysis by Metals. Clarendon Press, Oxford. carbons. J. chem. Phys. 36, 3401-3411. van Genuchten M. Th. and Cleary R. W. (1982) Movement Nkedi-Kizza P., Rao P. S. C. and Hornsby A. G. (1985) of solutes in soil: computer-simulated and laboratory Influence of organic cosolvents on sorption of hydro- results. In Soil Chemistry, Physico-Chemical Models phobic organic chemicals by soils. Envir. Sci. Techol. 19, (Edited by Bolt G. H.). Elsevier Science, New York. 975-979. van Genuchten M. Th. and Wierenga P, J. (1976) Mass Parker J. C. and Valocchi A. J. (1986) Constraints on transfer studies is sorbing porous media I. Analytical the validity of equilibrium and first-order kinetic trans- solutions. Soil Sci. Soc. Am. J. 40, 473-480. port models in structured soils. Wat. Resour. Res. 22, Giddings J. C. (1965) Dynamics of Chromatography. 399-407. Dekker, New York. Pinder G. F. (1984) Groundwater contaminant transport Glueckauf E. and Coates J. I. (1947) Theory of chromatog- modeling. Envir. Sci. Technol. 18, 108A-I14A. raphy. Part IV. The influence of incomplete equilibrium Radke C. J. and Prausnitz J. M. (I 972) Thermodynamics of on the front boundary of chromatograms and on the multi-solute adsorption from dilute liquid solutions. Am. effectiveness of separation. J. Chem. Soc. 1947, Inst. Chem. Engng J. 18, 761-768. 1315-1321. Rao P. S. C., Jessup R. E., Rolston D. E., Davidson J. M. Goodknight R. C., Klikoff W. A. Jr and Fatt I. (1960) and Kilcrease D. P. (1980a) Experimental and mathemati- Non-steady-state fluid flow and diffusion in porous media cal description of nonadsorbed solute transfer by diffu- containing dead-end pore volume. J. phys. Chem. 64, sion in spherical aggregates. Soil Sci. Soc. Am. J. 44, 1162-I 168. 684-688. Grahame D. C. (1947) The electrical double layer and the Rao P. S. C., Rolston D. E., Jessup R. E. and Davidson theory of electrocapillarity. Chem. Rev. 41, 441-501. J. M. (1980b) Solute transport in aggregated porous Hayes K. F. and Leckie J. O. (1987) Modeling ionic strength media: theoretical and experimental evaluation. Soil Sci. effects on cation adsorption at hydrous oxide/solution Soc. Am. J. 44, 1139-1146. interfaces. J. Colloid Interface Sci. 115, 564-572. Roberts P. V., Goltz M. N., Summers R. S., Crittenden J. C. Haymaker J. W. and Thompson J. M. (1972) Adsorption. and Nkedi-Kizza P. (1987) The influence of mass transfer In Organic Chemicals in the Soil Environment (Edited by on solute transport in column experiments with an aggre- Goring C. M. and Haymaker J. M.), pp. 49-143. Dekker, gated soil. J. contain. Hydrol. I, 375-393. New York. Schwarzenbach R. P. and Westall J. (1981) Transport of Hiemenz P. C. (1986) Principles of Colloid and Surface nonpolar organic compounds from surface water to Chemistry. Dekker, New York. groundwater, laboratory soprtion studies. Envir. Sci. Hutzler N. J., Crittenden J, C,, Oravitz J. L., Meyer C. J. Technol. 15, 1360-1367. and Johnson A. S. (1984) Modeling groundwater trans- Sposito G. (1984a) The Chemistry of Soils. Oxford Univer- port of organic compounds. In Proceedings Environmental sity Press, New York. Engineers Speciality Conference. ASCE, Los Angeles, Sposito G. (1984b) Chemical models of inorganic pollutants Calif. in soils. CRC Crit. Rev. envir. Control 15, 1-24. Israelachvili J. N. (1985) Intermolecular and Surface Forces. Stumm W. and Morgan J. J. (1981) Aquatic Chemistry. Academic Press, London. Wiley, New York. James R. O. and Parks G. A. (1982) Characterization of Travis C. C. and Etnier E. L. (1981) A survey of sorptive aqueous colloids by their electrical double-layer and relationships for reactive solutes in soil. J. envir. Qual. 10, intrinsic surface chemical properties. Surf Colloid Sci. 12, 8-17. 119-217. Weber W. J. Jr (1972) Physicochemical Processes for Water Karickhoff S. W. (1984) Organic pollutant sorption in Quality Control. Wiley, New York. aquatic systems. J. Hyd. Engng Div. Am. Soc. civ. Engrs Weber W. J. Jr (1984) Modeling of adsorption and 110, 707-735. mass transport processes in fixed-bed adsorbers. In 528 WALTER J. WEBER JR et al.

Proceedings, First International Conference on the Funda- Weber W. J. Jr and Smith E. H. (1988) Simulation and mentals of Adsorption, Schloss Elmau, West Germany, design models for adsorption processes. Envir. Sci. Tech- May 1983 (Edited by Myers A. L. and Belfort G.). nol. 21, 1040-1050. Engineering Foundation and the American Institute of Westall J. and Hohl H. 0980) A comparison of electrostatic Chemical Engineers, New York. models for the oxidc/solution interface. Adv. Colloid Weber W. J. Jr and Crittenden J. C. (1975) MADAM I--a Interface Sci. 12, 265-294. numeric method for design of adsorption systems. J. Wat. Wu S. and Gsehwvnd P. M. (1986) Sorption kinetics of Pollut. Control Fed. 47, 924-940. hydrophobic organic compounds to natural sediments Weber W. J. Jr and Ruraer R. R. Jr (1965) Intraparticle and soils. Envir. Sci. Technol. 20, 717-725. transport of sulfonated alykbenzenes in a porous solid: Yalkowsky S. H. and Valvani S. C. (1980) Solubility diffusion with non-linear adsorption. Wat. Resour. Res. 1, and partitioning I: solubility of nonelectrolytes in water. 361-362. J. pharmac. Sci. 69, 912-922.