Nonlinear Sorption Isotherms Research Articles

Diffusion anomaly and blind pore character in carbon molecular sieve membrane

The structure of a carbon molecular sieve (CMS) membrane is characterized by the through pores and blind pores with non-linear sorption isotherms inside. Time-lag analysis was conducted for gas permeation in such a structure and mathematical formulations were derived for two cases. It is found that the pressure dependence of the time lag is dominated by the ratio of sorption affinities in the two types of pores. The experimental permeation data of pure component CO 2 and N 2 measured on a CMS membrane were used to validate the model. It is found that, for the adsorbing species ( CO 2 ) , the model is able to well describe the diffusion anomalies over a wide range of permeation pressure, while for the weakly adsorbing species ( N 2 ) , the model is inadequate to cope with the anomalies at the low end of permeation pressure.

Modeling Contaminant Transport in a Three-Phase Groundwater System with the Freundlich-Type Retardation Factor

Colloid-facilitated contaminant transport was simulated in this study for the three-phase groundwater system where one or more sorption processes can be described with nonlinear sorption isotherm (Freundlich isotherm). A concise form of contaminant transport equation was derived from the mass balance equation of the contaminant. The developed model was numerically solved by the finite difference method along with the Picard iteration. The simulation results were used to quantitatively analyze the previously reported column data showing nonlinear sorption behavior. The analysis led to the following observations: (i) increases of the distribution coefficient of contaminant between the aqueous and solid phases (Ks c) and the one between the dissolved natural organic matters and solid phase (Ks OM) generate less facilitation (i.e., late arrival of contaminant breakthrough curves (BTCs), and the distribution coefficient of contaminant between the aqueous and the solid phases (KOM c) gives the opposite result; (ii) the increase of the Freundlich constant for the sorption isotherm between the aqueous and the solid phases (Ns c) yields the late arrival of BTC, and the other two Freundlich constants produce the opposite results; (iii) the Freundlich constants generally yield a sharper front as the BTC arrives at later pore volumes, while the distribution coefficients generally yield a more spread of the BTC as it arrives at later volumes. This modeling study shows that transport modeling provides a more efficient analyzing tool than the retardation factor alone concerning the colloid-facilitated contaminant transport with nonlinear sorption processes.

Sorption of male hormones by soils and sediments

This paper reports the sorption of two male hormones, testosterone and androstenedione, by four soil and sediment samples at both equilibrium and rate-limiting conditions. Unlike prior studies, androstenedione was studied independently of testosterone. Apparent sorption equilibrium is achieved in one to two weeks when the initial aqueous hormone concentrations (C0) at 10,000 microg/L (approximately 30% of their solubility limits [S(w)]) and two to three weeks when the C(0) is 300 microg/L (less than 1% of S(w)). The Freundlich model parameter n ranged from 0.698 to 0.899 for all soil-solute systems indicating nonlinear sorption isotherms. Isotherm nonlinearity leads to an inverse correlation between single-point organic carbon-normalized sorption distribution coefficients (K(oc)) and equilibrium androgen concentration (C(e)). When C(e)S(w) = 0.012, the log K(oc) values for testosterone and androstenedione on the various sorbents ranged from 6.18 to 6.75 and 6.83 to 6.04, respectively, compared to 6.30 to 6.80 and 6.16 to 6.92 when C(e)/S(w) = 0.004. This study suggests that male hormones may exhibit slow rates of sorption over 14 d or longer and that soils and sediments may have greater sorption distribution coefficients when concentrations fall into the ng/L range.

The effect of applied stresses on the equilibrium moisture content in polymers

The relation between saturation moisture content in a polymer and applied stresses is derived as a function of the coefficient of moisture expansion. The model is applied to the equilibrium water uptake of an isotropic polymer matrix composite . It is shown that a composite has a nonlinear sorption isotherm even though the matrix has a linear one, because of the internal stresses induced by swelling.

Humic Substance Adsorptive Fractionation by Minerals and its Subsequent Effects on Pyrene Sorption Isotherms

Changes in the nonlinearity of pyrene sorption isotherms on humic substance (HS)-coated minerals (kaolinite and hematite) due to HS adsorptive fractionation processes were examined in model environmental systems at low mass fraction organic carbon (foc) levels (0.0001–0.0011) using purified Aldrich humic acid (PAHA) and Suwannee River fulvic acid (SRFA). At a constant pH of 7, higher molecular weight (MW) fractions of PAHA were preferentially adsorbed on kaolinite whereas no adsorptive fractionation of PAHA occurred on hematite. At a constant foc level of 0.0005, preferential adsorption of higher MW PAHA fractions on kaolinite was enhanced with increasing pH. Nonlinear pyrene sorption isotherms were observed with the bulk PAHA-coated mineral systems, whereas more linear pyrene sorption isotherms were observed for the PAHA-mineral systems undergoing adsorptive fractionation. Although the degree of isotherm linearity may be affected by pH and/or structural rearrangement of the adsorbed HS fractions on minerals, this study suggests that HS adsorptive fractionation is more important than are changes in pH and foc levels with regard to the resulting pyrene sorption isotherms. Similar effects were not observed with SRFA, suggesting that the impacts of HS adsorptive fractionation on pyrene sorption isotherm nonlinearity are also influenced by the source and other biogeochemical characteristics of HS.

Chemical Extractions Affect the Structure and Phenanthrene Sorption of Soil Humin

Humin is a major fraction of soil organic matter and strongly affects the sorption behavior and fate of organic contaminants in soils and sediments. This study evaluated four different extraction methods for soil humins in terms of their organic carbon structural changes and the consequent effects on phenanthrene sorption. Solid-state 13C NMR demonstrated that 0.1 M NaOH exhaustively extracted humin and humin extracted with 6 M HF/HCl at 60 degrees C had a relatively high amount of aliphatic components as compared with 1 M HF-extracted humin. The treatment of 6 M HF/HCl at 60 degrees C greatly reduced carbohydrate components (50-108 ppm) from humin samples, i.e., more than 50% reduction. In addition, the humin from this 6 M HF/HCl treatment contained relatively more amorphous poly(methylene) domains than the humins extracted by other methods. With the respect to phenanthrene sorption, the linearity of sorption isotherm (N) and sorption affinity (Koc) varied markedly among the humin samples extracted by different methods. The NaOH exhaustively extracted humin had the most nonlinear sorption isotherm and the HF-extracted humin had the lowest Koc. It is concluded that humin samples from different extraction procedures exhibited substantial differences in their organic carbon structure and sorption characteristics, even though they were from the same soil. Therefore, one needs to be cautious when comparing the structural and sorption features of soil humins, especially when they are extracted differently. The 6 M HCl/HF extraction at elevated temperature is not encouraged, due to the modifications of chemical structure and physical conformation of organic matter.

Gas permeation in hollow fiber membranes with nonlinear sorption isotherm and concentration dependent diffusion coefficient

Gas permeation in membranes of cylindrical geometry (hollow fiber) is studied. Mathematical formulation is derived for the time-lag method, assuming Langmuir sorption equilibrium and Darken-type diffusion. Analytical expressions for the steady state concentration profile and the rate of permeation are also provided. It is demonstrated that the use of the traditional time-lag expression for a slab ( L t = l s 2 / 6 D ) in the place of a hollow fiber may incur large errors for both adsorbing and weakly adsorbing gases.

Phenanthrene and Pyrene Sorption and Intraparticle Diffusion in Polyoxymethylene, Coke, and Activated Carbon

We report sorption isotherms and uptake kinetics for phenanthrene and pyrene with three organic model sorbents: polyoxymethylene (POM), coke, and activated carbon. We combine batch equilibration and kinetic experiments with the direct observation of the long-term diffusion of phenanthrene and pyrene as measured within cross-sectioned particles using microprobe laser-desorption laser-ionization mass spectroscopy (muL2MS). For POM pellets, the intraparticle concentration profiles predicted from kinetic batch experiments and a polymer diffusion model with spherical geometry are in agreement with the independent muL2MS measurements. For coke particles, the apparent diffusivities decreased with smaller particle size. These trends in diffusivities were described by a sorption-retarded pore diffusion model with a particle-size-dependent solid-water partitioning coefficient obtained from apparent equilibrium observed in the kinetic batch studies. For activated carbon, the muL2MS measurements showed faster radial diffusion of phenanthrene and pyrene into the particle interior than predicted from diffusion models based on a single sorption domain and diffusivity. A branched pore kinetic model, comprising polycyclic aromatic hydrocarbon (PAH) macropore diffusion with kinetic exchange of PAH between macroporous and microporous domains, fits the experimental observations better. Because of parallel macro- and microdiffusion processes, nonlinear sorption isotherms, and a concentration-dependent diffusivity, it is not possible to make independent parameter estimations for intraparticle diffusion in activated carbon using our present procedures.

The influence of the rigidity of geosorbent organic matter on non-ideal sorption behaviors of chlorinated benzenes

This study focused on evaluating the influence of the rigidity of natural organic matter (NOM) associated with four natural geosorbents in controlling the non-ideal sorption behaviors of five chlorinated benzenes. Single solute sorption isotherms for each sorbate/sorbent combination were modeled and interpreted by the Freundlich sorption isotherm and the adsorption-partitioning model based on Polanyi–Manes theory (PM model). “Rigid” organic matter was operationally quantified as the fraction of carbon resistant to wet chemical oxidation (hard carbon) or thermal oxidation (soot carbon); atomic H/O ratios indicated a close correlation between the degree of reduction of the NOM and its rigidity. Sorbents with larger rigid carbon fractions had more non-linear sorption isotherms and higher organic carbon (OC) normalized sorption affinities. The size of the PM hole filling domain for a given sorbent was closely correlated with the extent to which the sorbent's affinity for chlorobenzenes exceeded predictions from a linear free energy relationship. Loss of some portions of the rigid character of the NOM domain due to the penetration of sorbate molecules (plasticization) was discussed as a possible contributor to the non-ideal sorption behaviors observed in this study. The existence of a permanently rigid NOM domain, not subject to plasticization under environmental conditions, was postulated as an additional factor determining the observed sorption behavior.

Correlations of nonlinear sorption of organic solutes with soil/sediment physicochemical properties

The estimation of solute sorptive behaviors is essential when direct sorption data are unavailable and will provide a convenient way to assess the fate and the biological activity of organic solutes in soil/sediment environments. In this study, the sorption of 2,4-dichlorophenol (2,4-DCP) on 19 soil/sediment samples and the sorption of 13 organic solutes on one sediment were investigated. All sorption isotherms are nonlinear and can be described satisfactorily by a simple dual-mode model (DMM): q e = K p C e + Q 0 · bC e/(1 + bC e), where K p (ml g −1) is the partition coefficient; C e (μg ml −1) is the equilibrium concentration; Q 0 (μg g −1) is the maximum adsorption capacity; Q 0 · b (ml g −1) is the Langmuir-type isotherm slope in the low concentration (Henry’s law) range and b (ml μg −1) is a constant related to the affinity of the surface for the solute. Based on these nonlinear sorption isotherms and similar other nonlinear isotherms, it is observed that, for both polar 2,4-DCP and nonpolar phenanthrene, K p, Q 0 and Q 0 · b are linearly correlated with soil/sediment organic carbon content ( f oc in the range of 0.118–53.7%). The results indicate that the nonlinear sorption of organic solutes results primarily from interactions with soil/sediment organic matter. The K oc ∗ ( K oc ∗ = K p / f oc ) , Q oc ( Q oc = Q 0/ f oc), L oc ( L oc = Q 0 · b/ f oc) and b for a given organic solute with different soils/sediments are largely invariant. Furthermore, log K oc ∗ , log b and log L oc for various organic solutes are correlated significantly with the solute log K ow or log S w ( log K ow in the range of 0.9 to 5.13 and log S w in the range of −6.176 to −0.070). A fundamental empirical equation was then established to calculate approximately the nonlinear sorption from soil/sediment f oc and solute S w for a given solute equilibrium concentration.

Desorption kinetics of phenanthrene in aquifer material lacks hysteresis.

Desorption experiments were carried out in flow through columns following long-term sorption batch experiments (up to 1010 days at 20 degrees C; Rügner, H.; Kleineidam, S.; Grathwohl, P. Long-term sorption kinetics of phenanthrene in aquifer materials. Environ. Sci. Technol. 1999, 33, 1645-1651) to elucidate sorption/desorption hysteresis phenomena of phenanthrene in aquifer materials. Most of the sorbents employed in this study (homogeneous lithocomponents separated from aquifer sediments or fresh rock fragments) showed highly nonlinear sorption isotherms because of coal particles embedded inside the grains. Because sorption capacities were high, sorption equilibrium was not reached in most of the sorbents during the initial sorptive uptake experiments lasting up to 1010 days. Desorption was studied up to 90 days at 20 degrees C. The temperature was raised after that stepwise from originally 20 to 30, 40, 50, and finally to 70 degrees C for selected samples to estimate activation energies of desorption. A numerical intraparticle pore diffusion model was used to fit sorptive uptake data and subsequently for pure forward prediction of the release rates in the desorption column experiments. Desorption was initially fast followed by extended tailing which in other studies is fitted by using multirate first-order models. Our results demonstrate that the retarded intraparticle pore diffusion model can predict the desorption rates with a single diffusion rate constant obtained independently from the long-term batch sorption experiment. No evidence for hysteresis was found, suggesting that many hysteresis phenomena reported earlier are experimental artifacts resulting from nonequilibrium effects and "nonphysical" models. The different temperature steps allowed one to additionally calculate activation energies of desorption (45-59 kJ mol(-1)), which were in reasonably good agreement with results from earlier studies for a retarded pore diffusion process. In addition, equilibrium sorption isotherms were determined at 20 and 40 degrees C to compare sorption and desorption enthalpies. Both were in good agreement, confirming that desorption was not significantly different from sorption.

Herbicide Runoff along Highways. 2. Sorption Control

Controversy remains about the importance of nonlinear sorption isotherms, desorption rate limitations, and aging effects, collectively referred to as nonideal sorption processes, in controlling the fate and transport of organic contaminants. Herbicide runoff from highway soils represents a good test case for assessing the relative importance of nonideal sorption because runoff flow rates are often high, soil-water contacttimes are short, and significant time is available for contaminant aging after application. This study examines the sorption and desorption of five herbicides with a wide range of properties (isoxaben, oryzalin, diuron, clopyralid, and glyphosate) on soil samples from two roadsides in northern California and uses the results to examine field runoff data from multiple rainy seasons. Nonideal sorption processes do not appear to be significant in determining herbicide runoff at the field sites because (i) sorption isotherms were linear or slightly nonlinear for all compounds but glyphosate, (ii) field runoff concentration ratios between isoxaben and oryzalin were consistent with linear partitioning predictions, (iii) runoff leaving the site appeared to be in equilibrium with local soil concentrations, and (iv) desorption distribution coefficients for aged herbicides on soil samples collected from the field site did not differ substantially from those obtained in short-term laboratory adsorption experiments. Collectively, these findings indicate that linear equilibrium models are adequate for predicting the concentration of herbicides in runoff in these field settings and that more complicated nonideal models do not need to be invoked. Vegetated slopes effectively reduced the herbicide loads, with average removals of 35-80% occurring as runoff traversed a 3-m segment 1 m from the edge of the spray zone.

Pyrene Sorption to Water-Dispersible Colloids: Effect of Solution Chemistry and Organic Matter

Chemical sorption to mobile soil colloids is a controlling factor for colloid-facilitated chemical transport in the vadose zone and groundwater. We investigated sorption of pyrene to soil colloid suspensions originating from soils differing in organic matter content for different solution chemistries. Colloids were obtained from two soils with different organic matter contents but similar geological histories by three different methods: (i) chemical dispersion, (ii) mechanical dispersion in water, and (iii) spontaneous release in water. Batch sorption experiments were conducted at five pyrene concentrations, in either pure water or at two different concentrations of K + and Ca 2+ . Generally, K + addition enhanced pyrene sorption, whereas Ca 2+ addition decreased sorption. The chemically dispersed colloids exhibited the highest pyrene sorption capacity and had the most nonlinear sorption isotherms, whereas whole soil had the most linear isotherm. Model calculations of the potential amounts of leachable pyrene illustrated the importance of including both colloid- and dissolved organic matter (DOM)-facilitated transport in risk assessment models when dealing with pyrene transport. The leaching potential of dissolved pyrene (with no DOM- and colloid-facilitated transport) was 5% of the leaching potential when both DOM- and colloid-sorbed pyrene was included.

Pyrene Sorption to Water‐Dispersible Colloids: Effect of Solution Chemistry and Organic Matter

Chemical sorption to mobile soil colloids is a controlling factor for colloid‐facilitated chemical transport in the vadose zone and groundwater. We investigated sorption of pyrene to soil colloid suspensions originating from soils differing in organic matter content for different solution chemistries. Colloids were obtained from two soils with different organic matter contents but similar geological histories by three different methods: (i) chemical dispersion, (ii) mechanical dispersion in water, and (iii) spontaneous release in water. Batch sorption experiments were conducted at five pyrene concentrations, in either pure water or at two different concentrations of K+ and Ca2+ Generally, K+ addition enhanced pyrene sorption, whereas Ca2+ addition decreased sorption. The chemically dispersed colloids exhibited the highest pyrene sorption capacity and had the most nonlinear sorption isotherms, whereas whole soil had the most linear isotherm. Model calculations of the potential amounts of leachable pyrene illustrated the importance of including both colloid‐ and dissolved organic matter (DOM)‐facilitated transport in risk assessment models when dealing with pyrene transport. The leaching potential of dissolved pyrene (with no DOM‐ and colloid‐facilitated transport) was 5% of the leaching potential when both DOM‐ and colloid‐sorbed pyrene was included.

Competitive sorption of organic contaminants in chalk

In the Negev desert, Israel, a chemical industrial complex is located over fractured Eocene chalk formations where transfer of water and solutes between fracture voids and matrix pores affects migration of contaminants in the fractures due to diffusion into the chalk matrix. This study tests sorption and sorption competition between contaminants in the chalk matrix to make it possible to evaluate the potential for contaminant attenuation during transport in fractures. Single solute sorption isotherms on chalk matrix material for five common contaminants ( m-xylene, ametryn, 1,2-dichloroethane, phenanthrene, and 2,4,6-tribromophenol) were found to be nonlinear, as confirmed in plots of K d versus initial solution concentration. Over the studied concentration ranges, m-xylene K d varied by more than a factor of 100, ametryn K d by a factor of 4, 1,2-dichloroethane K d by more than a factor of 3, phenanthrene K d by about a factor of 2, and 2,4,6-tribromophenol K d by a factor of 10. It was earlier found that sorption is to the organic matter component of the chalk matrix and not to the mineral phases (Chemosphere 44 (2001) 1121). Nonlinear sorption isotherms indicate that there is at least some finite sorption domain. Bi-solute competition experiments with 2,4,6-tribromophenol as the competitor were designed to explore the nature of the finite sorption domain. All of the isotherms in the bi-solute experiments are more linear than in the single solute experiments, as confirmed by smaller variations in K d as a function of initial solution concentration. For both m-xylene and ametryn, there is a small nonlinear component or domain that was apparently not susceptible to competition by 2,4,6-tribromophenol. The nonlinear sorption domain(s) is best expressed at low solution concentrations. Inert-solvent-normalized single and bi-solute sorption isotherms demonstrate that ametryn undergoes specific force interactions with the chalk sorbent. The volume percent of phenanthrene sorbed at the liquid solubility limit is calculated to be 13% v:v in both the single and bi-solute experiments. This value exceeds what may be reasonably interpreted as partitioning of phenanthrene into organic matter, despite the relative linearity of the phenanthrene sorption isotherm (compared with other compounds) in both single and bi-solute systems.

Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform media

A new reactive transport modelling approach and examples of its application are presented, dealing with the impact of sorption/desorption kinetics on the spreading of solutes, e.g. organic contaminants, in groundwater. Slow sorption/desorption is known from the literature to be strongly responsible for the retardation of organic contaminants. The modelling concept applied in this paper quantifies sorption/desorption kinetics by an intra-particle diffusion approach. According to this idea, solute uptake by or release from the aquifer material is modelled at small scale by a “slow” diffusion process where the diffusion coefficient is reduced as compared to the aqueous diffusion coefficient due to (i) the size and shape of intra-particle pores and (ii) retarded transport of solutes within intra-particle pores governed by a nonlinear sorption isotherm. This process-based concept has the advantage of requiring only measurable model parameters, thus avoiding fitting parameters like first-order rate coefficients. In addition, the approach presented here allows for modelling of slow sorption/desorption in lithologically nonuniform media. Therefore, it accounts for well-known experimental findings indicating that sorptive properties depend on (i) the grain size distribution of the aquifer material and (ii) the lithological composition (e.g. percentage of quartz, sandstone, limestone, etc.) of each grain size fraction. The small-scale physico-chemical model describing sorption/desorption is coupled to a large-scale model of groundwater flow and solute transport. Consequently, hydraulic heterogeneities may also be considered by the overall model. This coupling is regarded as an essential prerequisite for simulating field-scale scenarios which will be addressed by a forthcoming publication. This paper focuses on mathematical model formulation, implementation of the numerical code and lab-scale model applications highlighting the sorption and desorption behavior of an organic contaminant (Phenanthrene) with regard to three lithocomponents exhibiting different sorptive properties. In particular, it is shown that breakthrough curves (BTCs) for lithologically nonuniform media cannot be obtained via simple arithmetic averaging of breakthrough curves for lithologically uniform media. In addition, as no analytical solutions are available for model validation purposes, simulation results are compared to measurements from lab-scale column experiments. The model results indicate that the new code can be regarded as a valuable tool for predicting long-term contaminant uptake or release, which may last for several hundreds of years for some lithocomponents. In particular, breakthrough curves simulated by pure forward modelling reproduce experimental data much better than a calibrated standard first-order kinetics reactive transport model, thus indicating that the new approach is of high quality and may be advantageously used for supporting the design of remediation strategies at contaminated sites where some lithocomponents and/or grain size classes may provide a long-term pollutant source.

Permeation Time Lag and the Concentration Dependence of the Diffusion Coefficient of CO2 in a Carbon Molecular Sieve Membrane

The CO2 permeation experiments were performed on a microporous carbon molecular sieve membrane (CMSM) at 25 °C and over a wide pressure range. The permeation time lag of the penetrant was employed to study the concentration dependence of the diffusion coefficient. The analytical solutions are derived for permeation systems with a nonlinear sorption isotherm (Langmuir) and a concentration-dependent diffusivity [with the functional form of Dμ(c) = 1/(1 − θ)n, where θ is the ratio of surface coverage and n ≥ 1]. By comparison of the experimental data with the theoretical models, it is found that, in this CMSM, the apparent diffusivity of CO2 takes a concentration dependence stronger than that described by the traditional HIO model [i.e., Dμ(c)/ = 1/(1 − θ)].

Isotope effects in α-PdH(D) as an instrument for diagnosing bulk defects

Protium and deuterium sorption in the α-phase region is studied for highly defective palladium electrodeposits fabricated under controllable potentiostatic modes, particularly in the region of concentrated hydride formation. An anomalously high hydrogen content is observed for these samples in both α- and β-hydrides. On the basis of coulometry in the course of anodic hydrogen extraction, the non-linear sorption isotherms are plotted and their specific features are considered under an assumption of simultaneous equilibrium sorption of hydrogen by several types of lattice positions with a certain degree of defectiveness. Less defective palladium samples deposited in the absence of bulk hydridization are studied for comparison. The approach is proposed to estimate the fraction of defective regions. The procedure of analyzing the low-pressure limit is used for the first time for determining the specific values of the isotope effect, and also H and D Sievert's constants. The isotope effect is demonstrated to be extremely sensitive to the type of defectiveness.

Solute transport under non-linear sorption and decay

Contaminant transport in aquifers is usually represented by a solution to the advective–dispersive differential equation. When the contaminant is subject to non-linear degradation or decay, or it is characterized by a chemical constituent that follows a non-linear sorption isotherm, the resulting differential equation is non-linear. Using the method of decomposition, series solutions were obtained for the non-linear equation. The series were used to derive and test “simulant” solutions that arise using the concept of double decomposition. Simulant solutions are closed-form analytic expressions that approximate part of the series. These expressions are simple, stable, and flexible. They permit an accurate forecasting of contaminant propagation under non-linearity in laboratory or field investigations at early or prolonged times after the spill. In this article, the practical scenario of an instantaneous spill, and that of a constant concentration boundary condition, is studied for situations of non-linear decay, non-linear Freundlich isotherm, and non-linear Langmuir isotherm. The solutions are verified with limited well-known analytical solutions of the linear reactive and non-reactive equations with excellent agreement, and with limited finite difference solutions. Plumes undergoing non-linear decay experience a profile re-scaling with respect to that of linear decay, the degree of which is controlled by the magnitude of the non-linear parameter b. The direction of the scaling (scaling up or scaling down with respect to the linear decay plume) is controlled by the magnitude of C (whether greater or less than 1) in relation to the magnitude of b (whether greater or less than 1). When C>1, values of b<1 produce plumes that experience less decay (i.e., are scaled up) than that of the linear decay, whereas values of b>1 produce non-linear plumes that experience more decay (i.e., are scaled down) than that of the linear decay. The opposite effect is observed when concentrations are less than 1. In other words, when C<1, values of b<1 produce non-linear plumes that experience more decay (i.e., are scaled down) than that of the linear decay, whereas values of b>1, produce non-linear plumes that experience less decay (i.e., are scaled up) than that of the linear decay. A plume undergoing non-linear sorption according to a Freundlich isotherm retards the processes of advection and dispersion with respect to a plume with no sorption. Similar to the case of non-linear decay, whether this retardation is more or less pronounced than that of the linear sorption plume depends on whether the values of b and C are greater or less than 1. The solution presented here for the advective dispersive equation subject to a Freundlich sorption isotherm is restricted to concentration greater than 1. When C>1 and b<1, the decrease in mobility in the non-linear plume is not as pronounced as that of a plume modeled by a linear isotherm. Plume shape may be quite sensitive to the values of the non-linear parameters. Plumes with parameter values b<1 (and C>1) exhibit the well-known lack of symmetry with respect to their center of mass, sharp fronts, and the tailing effects observed at hazardous waste sites. As the magnitude of the non-linear parameter increases, the non-linear plume approaches the linear one. This partial non-linear “retardation” can now be observed quantitatively with the models presented herein. The models developed also simulate the case of b >1 (i.e., “unfavorable” sorption), which produce a plume even more retarded than the linear. The shape of a contaminant plume following a non-linear Langmuir isotherm is very sensitive to the magnitude of the non-linear parameter α. Approximate solutions for mild non-linearity are presented.

Effect of mass-transfer limitations on bioavailability of sorbed naphthalene in synthetic model soil matrices

External and internal mass-transfer resistances influencing the bioavailability of sorbed naphthalene in a synthetic model matrix for soil aggregates were investigated in batch experiments in mixed reactors. Amberlite® adsorption resins (XAD4 and XAD7) were used as the synthetic model for soil aggregates. The effect of hydrodynamic conditions in the slurry phase on the diffusive transport across a stagnant film surrounding the model particles was studied. In addition, a mechanistic model was developed based on mass balances, diffusion equations, a nonlinear sorption isotherm, and microbial degradation kinetics. Experimental results could be explained well with this model. In the absence of external transfer limitations, intraparticle effective diffusion coefficients of (3.55 ± 0.10) × 10−9 m2/s and (5.29 ± 0.86) × 10−10 m2/s were determined for naphthalene in Amberlite XAD4 and XAD7, respectively.