Nonlinear Behavior of Heavy Metals in Soils Mobility and Bioavailability

H. Magdi Selim


Transport Equations 3

Retention Models 7

First-Order and Freundlich Models 7

Second-Order and Langmuir 10

Hysteresis 11

Multiple Reaction Models 14

Second-Order Models (SOMs) 19

Transport in Layered Soils 23

Concluding Remarks 34

References 34

The dynamics of heavy metals and their transport in the soil profile play a significant role in their bioavailability and leaching losses beyond the root-zone . The primary emphasis in this chapter is on different approaches that describe the dynamics of heavy metals in the soil system. Such knowledge is necessary because these approaches provide direct information on the concentration of heavy metals in the soil solution and thus on their dynamics and bioavailability in soils . Moreover, such predictive capability requires knowledge of the physical, chemical, as well as biological processes influencing heavy metal behavior in the soil environment

A number of theoretical models describing the transport of dissolved chemicals in soils have been proposed over the past three decades . One class of models deals with solute transport in well-defined geometrical systems where one assumes that the bulk of the solute moves in pores and/or cracks of regular shapes or through intra-aggregate voids of known geometries . Examples of such models include those dealing with the soil matrix of uniform spheres, rectangular or cylindrical voids, and discrete aggregate or spherical size geometries . Solutions of these models are analytic, often complicated, and involve several numerical approximation steps . In contrast, the second group of transport models consists of empirical models that do not consider well-defined geometries of the pore space or soil aggregates . Rather, solute transport is treated on a macroscopic basis with the water flow velocity, hydrodynamic dispersion, soil moisture content, and bulk density as the associated parameters that describe the soil system . Refinements of this macroscopic approach are the "mobile and immobile" transport models where local nonequilibrium conditions are due to diffusion or mass transfer of solutes between the mobile and immobile regions . Another refinement includes the "dual and multiporosity" type models, which are often referred to as two-flow or multiflow domain models . Others refer to such models as bimodal types, wherein the soil system can best be characterized by a bimodal pore size distribution . Mobile and immobile models as well as dual- and multi-flow have been used with various degrees of success to describe the transport of several solutes in soils These empirical or macroscopic models are widely used and far less complicated than the more exact approach for systems of well-defined porous media geometries mentioned above

To quantify the transport of heavy metals in the soil, models that include reactivity or retention and release reactions of the various heavy metal species with the soil matrix are needed Retention and release reactions in soils include ion exchange, adsorption-desorption, precipitation-dissolution, and other mechanisms such as chemical or biological transformations Retention and release reactions are influenced by several soil properties, including bulk density, soil texture, water flux, pH, organic matter, redox reactions, and type and amount of dominant clay minerals Adsorption is the process wherein solutes bind or adhere to soil matrix surfaces to form outer- or inner-sphere solute surface-site complexes In contrast, ion exchange reactions represent processes where charged solutes replace ions on soil particle surfaces Adsorption and ion exchange reactions are related in that an ionic solute species may form a surface complex and may replace another ionic solute species already on surface sites

Surface-complexation models have been used to describe an array of equilibrium-type chemical reactions, including proton dissociation, metal cation and anion adsorption reactions on oxides and clays, organic ligand adsorption, and competitive adsorption reactions on oxide and oxide-like surfaces The application and theoretical aspects of surface complexation models are extensively reviewed by Goldberg (1992) and Sparks (2003) . Surface-complexation models are chemical models based on a molecular description of the electric double layer using equilibrium-derived adsorption data They include the constant capacitance model, triple-layer model, and Stern variable surface charge models, among others Surface-complexation models have been incorporated into various chemical spe-ciation models . The MINEQL model was perhaps the first where the chemical speciation was added to the triple-layer surface-complexation model . Others include MINTEQ, SOILCHEM, HYDRAQL, MICROQL, and FITEQL (see Goldberg, 1992) .

All of the above-mentioned chemical equilibrium models require knowledge of the reactions involved and associated thermodynamic equilibrium constants Due to the heterogeneous nature of soils, extensive laboratory studies may be needed to determine these reactions Thus, predictions from transport models based on a surface-complexation approach may not describe heavy metal sorption by a complex soil system As a result, the need for direct measurements of the sorption and desorption and release behavior of heavy metals in soils is necessary Consequently, retention or the commonly used term "sorption" should be used when the mechanism of heavy metals removal from soil solution is not known, and the term "adsorption" should be reserved for describing the formation of solute-surface site complexes

In this chapter, several models that govern heavy metals transport and retention reactions in the soil are derived . Models of the equilibrium type are discussed first, followed by models of the kinetic type . Retention models of the multiple reaction type, including the two-site equilibrium-kinetic models, the concurrent and consecutive multireaction models, and the second-order approach are also discussed This is followed by multicomponent or competitive type models where ion exchange is considered the dominant retention mechanism Retention reactions of the reversible and irreversible types are incorporated into the transport formulation Selected experimental data sets will be described for the purpose of model evaluation and validation, and the necessary (input) parameters are discussed

Transport Equations

Dissolved chemicals present in the soil solution are susceptible to transport through the soil subject to the water flow constraints in the soil system. At any given point within the soil, the total amount of solute x (]g. cm-3) for a species i may be represented by

where S is the amount of solute retained by the soil (]g.g-1 soil), C is the solute concentration in solution (]g. mL-1), 8 is the soil moisture content (cm3 . cm-3), and p is the soil bulk density (g. cm-3) . The rate of change of x for the i-th species with time is subject to the law of mass conservation such that (omitting the subscript i)

or d(9C + pS) = _ (dJx + dJy + dJL) - Q dt dx dy dz

where t is time (h) and Jx, Jy, and Jz represent the flux or rate of movement of solute species i in the x-, y-, and z-directions (] .h-1), respectively. The term Q represents a sink (Q positive) or source that accounts for the rate of solute removal (or addition) irreversibly from the bulk solution (]g. cm-3 .h-1) . If we restrict our analysis to one-dimensional flow in the z-direction, the flux Jz, or simply J, in the soil may be given by

J =- 9 D + D }§ + qC (1.4)

where Dm is the molecular diffusion coefficient (cm2 .h-1), DL is the longitudinal dispersion coefficient (cm2 .h-1), and q is Darcy's flux (cm.h-1). Therefore, the primary mechanisms for solute movement are due to diffusion plus dispersion and by mass flow or convection with water as the water moves through the soil . The molecular diffusion mechanism is due to the random thermal motion of molecules in solution and is an active process regardless of whether or not there is net water flow in the soil . The result of the diffusion process is the well-known Fick's law of diffusion where solute flux is proportional to the concentration gradient The longitudinal dispersion term of Equation (1 .4) is due to the mechanical or hydrodynamic dispersion phenomena, which are due to the nonuniform flow velocity distribution during fluid flow in porous media. According to Fried and Combarnous (1971), nonuniform velocity distribution through the soil pores is a result of variations in pore diameters along the flow path, fluctuation of the flow path due to the tortuosity effect, and the variation in velocity from the center of a pore (maximum value) to zero at the solid surface interface (Poiseuille's law). The effect of dispersion is that of solute spreading, which is a tendency opposite to that of piston flow Dispersion is effective only during fluid flow, that is, for a static water condition or when water flow is near zero; molecular diffusion is the dominant process for solute transport in soils For multidimensional flow, longitudinal dispersion coefficients (DL) and transverse dispersion coefficients (DT) are needed to describe the dispersion mechanism . Longitudinal dispersion refers to that in the direction of water flow and that for the transverse directions for dispersion perpendicular to the direction of flow

Apparent dispersion D is often introduced to simplify the flux Equation (1 .4) such that

'-eDlC + 'C (1.5)

where D now refers to the combined influence of diffusion and hydrody-namic dispersion for dissolved chemicals in porous media . Incorporation of flux Equation (1 . 5) into the conservation of mass Equation (1 . 3) yields the following generalized form for solute transport in soils in one dimension:

d9C + dS _d_ dC dqC Q dt + P dt _ dz dz ' dz 'Q (1.6)

The above equation is commonly known as the convection-dispersion equation (CDE) for solute transport and is valid for soils under transient and unsaturated soil water flow conditions . For conditions where steady water flow is dominant, D and 8 are constants; that is, for uniform 8 in the soil, we have the simplified form of the CDE as follows:

dt 8 dt dz2 - V dz - 8 (1 . 7)

where v (cm.h-1) is commonly referred to as the pore-water velocity and is given by (q/8) .

Solutions of the above CDE Equations (1 .6) or (1 . 7) yield the concentration distribution of the amount of solute in soil solution C and that retained by the soil matrix S with time and depth in the soil profile . To arrive at such a solution, the appropriate initial and boundary conditions must be specified Several boundary conditions are identified with the problem of solute transport in porous media The simplest is that of a first-order type boundary condition such that a solute pulse input is described by

where Cs (]ug. cm-3) is the concentration of the solute species in the input pulse . The input pulse application is for a duration T, which was then followed by a pulse input that is free of such a solute Such a boundary condition was used by Lapidus and Amundson (1952) . The more precise third-type boundary condition at the soil surface was considered by Brenner (1962) in his classical work, where advection plus dispersion across the interface was considered. A continuous solute flux at the surface can be expressed as vCs = - D ^ + vC, dz z = O, t > O

and a flux-type pulse input as vCs = - D — + vC, z = O, t < T dz

The advantages of using third-type boundary conditions were discussed by Selim and Mansell (1976) and Kreft and Zuber (1978) . The boundary condition at some depth L in the soil profile is often expressed as (Danckwerts, 1953; Brenner, 1962; Lindstrom et al ., 1967; Kreft and Zuber, 1978)

which is used to deal with solute effluent from soils having finite lengths . However, it is often convenient to solve the CDE where a semi-infinite rather than a finite length (L) of the soil is assumed . Under such circumstances, the appropriate condition for a semi-infinite medium is f ^ 0 z ^ ' * 0 (1.14)

Analytical solutions to the CDE subject to the appropriate boundary and initial conditions are available for a number of situations, whereas the majority of solute transport problems must be solved using numerical approximation methods . In general, whenever the form of the retention reaction is a linear one, a closed-form solution is obtainable A number of closed-form solutions are available from Kreft and Zuber (1978), and Van Genuchten and Alves (1982) However, most retention mechanisms are nonlinear and time dependent in nature, and analytical solutions are not available As a result, a number of numerical models using finite-difference or finite element have been utilized to solve nonlinear retention problems of multireaction and multicomponent solute transport in soils (see Selim, Selim, Amacher, Iskandar, 1990) .

Retention Models

The form of heavy metal reactions in the soil system must be identified if prediction of their fate in the soil is sought. In general, heavy metals retention processes in soils have been quantified by scientists using several approaches . One approach represents equilibrium reactions and the second represents kinetic or time-dependent type reactions . Equilibrium models are those where heavy metal reaction is assumed to be fast or instantaneous in nature . Under such conditions, "apparent equilibrium" may be observed in a relatively short reaction time . Langmuir and Freundlich models are perhaps the most commonly used equilibrium models for the description of fertilizer chemicals such as phosphorus and for several heavy metals These equilibrium models include the linear and Freundlich (nonlinear) and the one- and two-site Langmuir type . Kinetic models represent slow reactions where the amount of solute sorption or transformation is a function of contact time . Most common is the first-order kinetic reversible reaction for describing time-dependent adsorption-desorption in soils . Others include linear irreversible and nonlinear reversible kinetic models . Recently, a combination of equilibrium and kinetic type (two-site) models, and consecutive and concurrent multireaction type models has been proposed

First-Order and Freundlich Models

The first-order kinetic approach is perhaps one of the earliest single forms of reactions used to describe sorption versus time for several dissolved chemicals in soils . This may be written as dS_ d t

where the parameters kf and kb represent the forward and backward rates of reactions (h-1) for the retention mechanism, respectively. The first-order reaction was originally incorporated into the classical CDE by Lapidus and Amundson (1952) to describe solute retention during transport under steady-state water flow conditions Integration of Equation (1 15) subject to initial conditions of C = Ci and S = 0 at t = 0, for several Ci values, yields a system of linear sorption isotherms . That is, for any reaction time t, a linear relation between S and C is obtained

Linear isotherms are not often encountered except for selected cations and heavy metals at low concentrations Isotherms that exhibit nonlinear or curve linear retention behavior are commonly observed for several reactive chemicals as depicted by the nonlinear isotherms for nickel and arsenic shown in Figures 1 .1, 1 .2, 1 .3 respectively (Liao and Selim, 2010; Zhang and Selim, 2005; Selim and Ma, 2001). To describe such nonlinear behavior,

figure 1.1

Adsorption isotherms for Ni on Webster soil at different retention times . The solid curves are based on the Freundlich equation.

figure 1.1

Adsorption isotherms for Ni on Webster soil at different retention times . The solid curves are based on the Freundlich equation.

Jk 400

Sorption Isotherms Windsor Soil

Sorption Isotherms Windsor Soil

Jk 400

10 20 30 40 50 Arsenic Concentration (mg/L)

10 20 30 40 50 Arsenic Concentration (mg/L)

figure 1.2

Adsorption isotherms for arsenic on Windsor soil at different retention times

"eg P

Sorption Isotherms McLaren Soil

Sorption Isotherms McLaren Soil

20 40 60

Cu Concentration (mg/L)

20 40 60

Cu Concentration (mg/L)

figure 1.3

Adsorption isotherms for copper on McLaren soil at different retention times .

the single reaction given in Equation (1 . 15) is commonly extended to include nonlinear kinetics such that (Selim and Amacher, 1997)

where b is a dimensionless parameter commonly less than unity, represents the order of the nonlinear or concentration-dependent reaction, and illustrates the extent of heterogeneity of the retention processes . This nonlinear reaction (Equation (1 .16)) is fully reversible and the magnitudes of the rate coefficients dictate the extent of kinetic behavior of retention of the solute from the soil solution. For small values of kf and kb, the rate of retention is slow and strong kinetic dependence is anticipated . In contrast, for large values of kf and kb, the retention reaction is a rapid one and should approach quasi-equilibrium in a relatively short time. In fact, at large times (t ^ 4), when the rate of retention approaches zero, Equation (1 . 16) yields

S = KfCh where Kf =

Equation (1 . 17) is analogous to the Freundlich equilibrium equation where Kf is the solute partitioning coefficient (cm3 . g-1) . Therefore, one may regard the parameter Kf as the ratio of the rate coefficients for sorption (forward reaction) to that for desorption or release (backward reaction) .

The parameter b is a measure of the extent of the heterogeneity of sorption sites of the soil matrix In other words, sorption sites have different affinities for heavy metal retention by matrix surfaces, where sorption by the highest energy sites takes place preferentially at the lowest solution concentrations For the simple case where b = 1, we have the linear form:

where the parameter Kd is the solute distribution coefficient (cm3 .g-1) and of similar form as the Freundlich parameter Kf . There are numerous examples of cations and heavy metals retention, which were described successfully using the linear or the Freundlich equation (Sparks, 1989; Buchter et al ., 1988) . The lack of nonlinear or concentration-dependent behavior of sorption patterns as indicated by the linear case of Equation (1 18) is indicative of the lack of heterogeneity of sorption-site energies For this special case, sorption-site energies for linear sorption processes of heavy metals are regarded as relatively homogeneous

Second-Order and Langmuir

An alternative to the above first- and n-th order models is that of the second-order kinetic approach Such an approach is commonly referred to as the Langmuir kinetic and has been used for the prediction of phosphorus retention (Van der Zee and Van Riemsdijk, 1986) and heavy metals (Selim and Amacher, 1997). Based on second-order formulation, it is assumed that the retention mechanisms are site specific, where the rate of reaction is a function of the solute concentration present in the soil solution phase (C) and the amount of available or unoccupied sites 9 (^g.gr1 soil), by the reversible process,

where kf and kb are the associated rate coefficients (h) and S is the total amount of solute retained by the soil matrix. As a result, the rate of solute retention may be expressed as

where ST (^g.gr1 soil) represents the total number of total sorption sites . As the sites become occupied by the retained solute, the number of vacant sites approaches zero (^^ 0) and the amount of solute retained by the soil approaches that of the total capacity of sites, that is, S ^ ST . Vacant specific sites are not strictly vacant. They are assumed occupied by hydrogen, hydroxyl, or by other specifically sorbed species . As t ^ 4, that is, when the reaction achieves local equilibrium, the rate of retention becomes

Upon further rearrangement, the second-order formulation, at equilibrium, obeys the widely recognized Langmuir isotherm equation:

where the parameter K [= 8kf/kbp] is now equivalent to ra in Equation (1 . 21) and represents the Langmiur equilibrium constant. Sorption-desorption studies showed that highly specific sorption mechanisms are responsible for solute retention at low concentrations The general view was that metal ions have a high affinity for sorption sites of oxide minerals surfaces in soils . In addition, these specific sites react slowly with reactive chemicals such as heavy metals and are weakly reversible


Adsorption-desorption results are presented as isotherms in the traditional manner in Figures 1 .4 and 1 .5 and clearly indicate considerable hysteresis for nickel and arsenic retention in two different soils, respectively (Liao and Selim, 2010; and Zhang and Selim, 2005) . Other examples for copper and zinc are shown in Figures 1 .6 and 1 . 7 (Selim and Ma, 2001; Zhao and Selim). As seen from the family of curves, desorption did not follow the same path (i e , nonsingularity) as the respective adsorption isotherm This nonsingularity or hysteresis may result from the failure to achieve equilibrium adsorption prior to desorption If adsorption as well as desorption were carried out for times sufficient for equilibrium to be attained, or the kinetic rate coefficients were sufficiently large, such hysteretic behavior would perhaps be minimized (Selim et al ., 1976). Such hysteretic behavior resulting from a discrepancy between adsorption and desorption isotherms was not surprising in view of the strong kinetic retention behavior of these heavy metals in soils Several studies indicated that observed hysteresis in batch experiments may be due to kinetic retention behavior and slow release and/or irreversible adsorption

0.00 0.01 0.02 0.03 0.04

Ni in Solution (mM)

0.00 0.01 0.02 0.03 0.04

Ni in Solution (mM)

figure 1.4

Adsorption and desorption isotherms illustrating hysteresis behavior of Ni retention on Webster soil .

figure 1.4

Adsorption and desorption isotherms illustrating hysteresis behavior of Ni retention on Webster soil .

As Concentration (mg/L)

figure 1.5

Adsorption and desorption isotherms illustrating hysteresis behavior of As retention in Sharkey clay soil .

Cu Concentration (mg/L)

figure 1.6

Adsorption and desorption isotherms illustrating hysteresis behavior of Cu retention in McLaren soil .

0 10 20 30 40 50 60

Zn Concentration (mg/L)

figure 1.7

Adsorption and desorption isotherms illustrating hysteresis behavior of Zn retention in Windsor soil

0 10 20 30 40 50 60

Zn Concentration (mg/L)

Windsor Soil

Windsor Soil conditions Adsorption-desorption isotherms indicate that the number of irreversible or nondesorbable phases increased with time of reaction . Heavy metals may be retained by heterogeneous type sites having a wide range of binding energies At low concentrations, binding may be irreversible The irreversible amount almost always increased with time It is suggested that hysteresis for heavy metals is probably due to extremely high energy bonding with organic matter and layer silicate surfaces The fraction of nondes-orbable solutes is often referred to as specifically sorbed Others suggested that several solutes were fixed in a nonexchangeable form, which resulted in a lack of reversibility as well as hysteretic behavior Moreover, several researchers reported that the magnitude of hysteresis increases with longer sorption incubation periods Increasing hysteretic behavior upon aging has been observed consistently for several pesticides and heavy metals

Hysteresis has also been observed in ion exchange reactions for several cations, where the exchange of one sorbed cation for another is not completely reversible, that is, the forward and reverse exchange reactions do not result in the same isotherms The hysteretic behavior of cation exchange is abundantly reported in the literature; a critical review of this literature was published by Verburg and Baveye (1994) . From a survey of the literature, they were able to categorize several elements into three categories The elements in each category were found to show hysteretic exchange between groups, but not within groups . They proposed that exchange reactions are most likely multistage kinetic processes in which the later rate-limiting processes are a result of physical transformation in the system (e.g., surface heterogeneity, swelling hysteresis, and formation of quasi-crystals) rather than simply a slow kinetic exchange process where there exists a unique thermodynamic relationship for forward and reverse reactions . While this may be true in some circumstances, an apparent (pseudo) hysteresis also can result from slow sorption and desorption reactions, that is, lack of equilibrium (Selim et al . 1976). Regardless of the different reasons for hysteresis, it is evident that kinetic models such as those proposed in this study need to be complemented by detailed information on the mechanism(s) responsible for the slow kinetic reaction(s)

Multiple Reaction Models

Several studies showed that the use of single reaction models, such as those described above, is not adequate because such models are of the equilibrium or kinetic type The failure of single reaction models is not surprising as they only describe the behavior of one species with no consideration for the simultaneous reactions of others in the soil system Multicomponent models consider a number of processes governing several species, including ion exchange, complexation, precipitation and dissolution, and competitive adsorption, among others . Multicomponent models rely on the basic assumption of local equilibrium of the governing reactions where possible kinetic reactions are ignored .

Multisite or multireaction models deal with the multiple interactions of one species in the soil environment . Such models are empirical in nature and based on the assumption that a fraction of the total sites are highly kinetic, whereas the remaining fraction of sites interact slowly or instantaneously with that in the soil solution (Selim et al ., 1976; Selim and Amacher, 1997). Nonlinear equilibrium (Freundlich) and first- or n-order kinetic reactions were the associated processes Such a two-site approach proved successful in describing observed extensive tailing of breakthrough results Amacher, Selim, and Iskandar (1988) developed a multireaction model that includes concurrent and concurrent-consecutive processes of the nonlinear kinetic type . The model was capable of describing the retention behavior of Cd and Cr(VI) with time for several soils . In addition, the model predicted that a fraction of these heavy metals was irreversibly retained by the soil A schematic representation of the multireaction model is shown in Figure 1 . 8. In this model we consider the solute to be present in the soil solution phase (C) and in four phases representing solute retained by the soil matrix as Se, S1, S2, S3, and Sirr . We further assume that Se, S1, and S2 are in direct contact with the solution phase and are governed by concurrent type reactions . Here we assume that Se is the amount of solute sorbed reversibly and is in equilibrium with C at all times The governing equilibrium retention and release mechanism was of the nonlinear Freundlich type, as discussed previously

The retention and release reactions associated with S1 and S2 were considered in direct contact with C and reversible processes of the (nonlinear) kinetic type govern their reactions:

S = Se + Sj + S2 + S3 + Sirr (1 . 23)

Multireaction Kinetic Model ki







figure 1.8

A schematic representation of the multireaction kinetic model .

where k1 to k4 are the associated rates coefficients (h-1) . These two phases (Sj and S2) may be regarded as the amounts sorbed on surfaces of soil particles and chemically bound to Al and Fe oxide surfaces or other types of surfaces, although it is not necessary to have a priori knowledge of the exact retention mechanisms for these reactions to be applicable . Moreover, these phases may be characterized by their kinetic sorption and release behavior to the soil solution and thus are susceptible to leaching in the soil . In addition, the primary difference between these two phases not only lies in the difference in their kinetic behavior, but also in the degree of nonlinearity as indicated by the parameters n and m. The multireaction model also considers irreversible solute removal via a retention sink term Q in order to account for irreversible reactions such as precipitation and dissolution, mineralization, and immobilization, among others . We expressed the sink term as a first-order kinetic process:

where kirr is the associated rate coefficient (h-1) .

The multireaction model also includes an additional retention phase (S3) that is governed by a consecutive reaction with S2 This phase represents the amount of solute strongly retained by the soil that reacts slowly and reversibly with S2 and may be a result of further rearrangements of the solute retained on matrix surfaces . Thus, inclusion of S3 in the model allows the description of the frequently observed very slow release of solute from the soil The reaction between S2 and S3 was considered to be of the kinetic firstorder type, that is,

S = k5 S2 - k6 S3 . 28)

where k5 and k6 (h-1) are the reaction rate coefficients . If a consecutive reaction is included in the model, then Equation (1 . 26) must be modified to incorporate the reversible reaction between S2 and S3 . As a result, the following equation dS

p-ST=k3 ecn + p(k4 + k5)s2 - pk6S3 (1 . 29)

must be used in place of Equation (1 . 26) . The above reactions are nonlinear in nature and represent initial-value problems that are typically solved based on numerical approximations . In addition, the above retention mechanisms were incorporated, in a separate model, into the classical convection-dispersion equation in order to predict solute retention as governed by the multireac-tion model during transport in soils (Zhang and Selim, 2005) .

The capability of the multireaction approach discussed above in describing experimental batch data for arsenic retention is shown by the solid curves of Figure 1 .9. The results and model predictions are given for the various initial concentrations (Ci). Overall, good model predictions were observed for the wide range of input concentrations values considered The multireaction model used here accounts for several interactions of the reactive solute species (As) within the soil system Specifically, the model assumes that a fraction of the total sites is highly kinetic whereas the remaining fraction interacts slowly or instantaneously with solute in the soil solution . As illustrated in Figure 1 . 8, the model also accounts for irreversible reactions of the concurrent (Sirr) and consecutive type (S3 ) . As a result, different versions of the multireaction model shown in Figure 1 . 8 represent different reactions from which one can deduce possible retention mechanisms

In our simulations, the multireaction model was fitted to arsenic versus time for all input concentrations (Ci) simultaneously. As a result, an overall set of model parameters for the appropriate rate coefficients, applicable for the entire data set, was achieved. The examples shown in Figure 1 .9 are for two different model versions . In the first version, the simulations in Figure 1 .9 (top), a kinetic phase (S1), and an irreversible phase (Sirr) were considered, where the necessary model parameters were n, k1, k2, and kirr. In the second version, the simulations in Figure 1 .9 (bottom), a kinetic phase (S2), as well as a consecutive irreversible reaction represented by (S3) were considered . The presence of a consecutive S3 phase may occur as a result of further surface rearrangement of the adsorbed phase (see Figure 1 .8). For this version, model parameters considered were m, k3, k4, and k5, where all other model parameters were set equal to zero . It is obvious from the simulations shown

MRM Model

Concurrent Irreversible Reaction

Windsor Soil c o U

MRM Model

Concurrent Irreversible Reaction

Windsor Soil

0 50 100 150 200 250 300 350 400 450 500 Time, Hours

Ci m 80

jm c

0 50 100 150 200 250 300 350 400 450 500 Time, Hours

747 Cargo Door

0 50 100 150 200 250 300 350 400 450 500 Time, Hours figure 1.9

Experimental results of As(V) concentration in soil solution for Windsor soil versus time for all Co's . Dashed curves were obtained using the MRM with concurrent (top figure) and consecutive (bottom figure) irreversible reactions .

figure 1.9

Experimental results of As(V) concentration in soil solution for Windsor soil versus time for all Co's . Dashed curves were obtained using the MRM with concurrent (top figure) and consecutive (bottom figure) irreversible reactions .

in Figure 1 .9 that a number of model versions were capable of producing indistinguishable simulations of the data Similar conclusions were made by Amacher, Selim, and Iskandar (1988) for Cd and Cr(VI) for several soils . They also stated that it was not possible to determine whether the irreversible reaction is concurrent or consecutive, as both model versions provided similar fit of their batch data . For the example shown in Figure 1 .9, the use of a consecutive irreversible reaction provided an improved fit of Cu retention over other model versions . This finding is based on goodness-of-fit (r2 and root mean square errors) as well as visual observation of measured data and model simulations

Second-Order Models (SOMs)

The basic assumption of the second-order modeling approach is that there exist at least two types of retention sites for heavy metals on soil matrix surfaces . Moreover, the primary difference between these two types of sites is based on the rate of the proposed kinetic retention reactions . Furthermore, the retention mechanisms are site specific, wherein the rate of reaction is a function of not only the solute concentration present in the soil solution phase, but also the number of available retention sites on matrix surfaces

The original second-order model (SOM) was first proposed by Selim and Amacher (1988) to describe Cr retention and transport in several soils . Here, two types of sites were considered: the first was of the equilibrium type and the second was kinetically controlled type sites . Moreover, Smax (]g.g-1 soil) was considered to represent the total retention capacity or total number of sites on matrix surfaces . It is also assumed that Smax is an intrinsic soil property that is time invariant Therefore, based on the two-site approach, the total adsorption sites are given by

where Smax is the adsorption maximum, and (Se)max and (Sk)max are the total amounts or adsorption maxima for equilibrium and kinetic type sites, respectively (]g.g-1 soil) . If f represents the fraction of equilibrium type sites (Se)max to the total sites, we thus have

(Se) max = f Smax and (Sk)max = ( 1 - f ) Smax (1 .31)

Assuming 9 and 9 as the vacant or available sites (]g.g-1 soil) for adsorption on equilibrium and kinetic type sites (Se and Sk), respectively, we have

= (Se Lax - Se = f S max Se

= (Sk )max - Sk = ( 1 - f ) Smax - Sk with the total available sites equal to 9 = q>e + q>k. As the sites become filled or occupied by the retained solute, the number of vacant sites approaches zero, (q>e + qj ® S . In the meantime, the amount of solute retained by the soil matrix approaches the total capacity or sorption maxima (Se +Sk) ® Smax .

The second-order approach was successfully used for Cr retention and transport predictions by Selim and Amacher (1988) and for Zn retention by Hinz, Buchter, and Selim (1992). This model was recently modified such that the total adsorption sites Smax were not partitioned between Se and Sk phases based on a fraction of sites (Selim and Amacher, 1997; Ma and Selim, 1998) Instead, it was assumed that the vacant sites were available to both types of Se and Sk . Therefore, f is no longer required and the amount of solute adsorbed on each type of sites is only determined by the rate coefficients associated with each type of site As a result, sites associated with equilibrium or instantaneous type reactions will compete for available sites prior to slow or kinetic type sites being filled . Perhaps such a mechanism is in line with observations where rapid (equilibrium type) sorption is encountered first, followed by slow types of retention reactions We are not aware of the use of this second-order approach to describe heavy metal retention kinetics and transport in soils

In the following analysis we followed an overall structure for the second-order formulation similar to that described for the multireaction approach of Figure 1 . 8 where three types of retention sites are considered with one equilibrium-type site (Se) and two kinetic-type sites, namely S1 and S2 Therefore, we have q now related to the sorption capacity (Smax) by

The governing retention reactions can be expressed as follows (Ma and Selim, 1998):

dt = [k3 9 C $ - k4 S2] " K5 S2 (1 . 36)

The units for Ke are cm3 .]g-1; k1 and k3 have a derived unit of cm3 .]g-1 .h-1; and k2, k4, k5, and kirr are assigned units of h-1 .

The input parameter Smax of the second-order model is a major parameter and represents the total sorption of sites . Smax, which is often used to characterize heavy metal sorption, can be quite misleading if the experimental data do not cover a sufficient range of solution concentration and if other conditions such as the amounts initially sorbed prevail (Houng and Lee, 1998) . In an arsenic adsorption study, Selim and Zhang (2007) used Smax in the SOM model based on average values as determined from the Langmuir isotherm equation . This is a simple approach to obtain Smax estimated when a direct measurement of the sorption capacity is not available . Figures 1 . 10, 1 . 11, and 1 .12 show simulated adsorption results using the SOM model based on two model versions for three different soils . Based on visual observations of the overall of fit of the model to the experimental data, SOM provided good overall predictions of the kinetic adsorption data for arsenic

The question arises whether SOM model improvements can be realized when one relaxes the assumption of the use of Langmuir Smax and utilizes parameter optimization to arrive at a best estimate of the rate coefficients (e . g., k1, k2, and kirr or k3, k4, and k5) as well as Smax . Based on these results, Selim and Zhang (2007) concluded that the use of Langmuir Smax as an input parameter provided good predictions of the adsorption results Moreover, the retention kinetics predictions for As(V) shown are in agreement with the biphasic arsenic adsorption behavior observed on several soil minerals (Fuller et al ., 1993; Raven et al ., 1998; Arai and Sparks, 2002) as well as whole soils (Elkhatib et al ., 1984; Carbonell-Barrachina et al ., 1996) over different time scales (minutes to months)

A comparison of the multireaction (MRM) and second-order two-site (SOTS) models for their capability to predict arsenic concentration with time is given in Figure 1 . 13 . Selim and Zhang (2007) found that several model versions fit the data equally well, but the sorption kinetics prediction capability varied among the soils investigated MRM was superior to SOM, and the use of irreversible reaction into the model formulations was essential They also found that incorporation of an equilibrium sorbed phase into the various model versions for As(V) predictions should be avoided

The success of the second-order approach in describing As(V) retention results is significant because, to our knowledge, the SOM formulation described in this chapter has not been applied to metalloid elements like As . Previous use of the second-order formulation, which included a partitioning of the sites, indicated that for Cr and Zn the rate coefficients were highly concentration dependent (Selim and Amacher, 1988; Hinz, Buchter, and Selim, 1992) Selim and Ma (2001) successfully utilized the SOM model

u 40

SOM Model with Concurrent Irreversible Reaction

Windsor Soil

0 50 100 150 200 250 300 350 400 450 500 Time, Hours

m 80

SOM Model with Consecutive Irreversible Reaction

Windsor Soil

100 150

200 250 300 Time, Hours

350 400 450 500

figure 1.10

Experimental results of As(V) concentration in soil solution for Windsor soil versus time for all Co's . Dashed curves were obtained using SOM with concurrent (top figure) and consecutive (bottom figure) irreversible reactions to describe Cu adsorption as well as desorption or release following sorption . They concluded that the use of consecutive irreversible reaction (k5 in Figure 1 . 1) provided improvements in the description of the kinetic sorption and desorption of Cu compared to the concurrent irreversible reaction (kirr) . This finding is contrary to that from this study for As adsorption for all three soils . Such contradictions are not easily explained and are thus subjects for future research u

SOM Model with Concurrent Irreversible Reaction

Olivier Soil

Continue reading here: Davidson 1976 Adsorption Freundlich Retardation

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