sorption isotherms: a review on physical bases, modeling...

Applied Geochemistry 22 (2007) 249–275

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AppliedGeochemistry

Review

Sorption isotherms: A review on physical bases, modelingand measurement

G. Limousin a,b,*, J.-P. Gaudet b, L. Charlet c, S. Szenknect a,V. Barthes a, M. Krimissa d

a Atomic Energy Commission, Tracers Technology Laboratory, 38054 Grenoble Cedex, Franceb Laboratoire d’etude des Transferts en Hydrologie et Environnement (CNRS-INPG-IRD-UJF), BP 53, 38041 Grenoble Cedex, France

c Laboratoire de Geophysique Interne et Techtonophysique (CNRS-IRD-LCPC-UJF-Universite de Savoie), BP 53,

38041 Grenoble Cedex, Franced Electricite de France, Division Recherche et Developpement, Laboratoire National d’Hydraulique et d’Environnement – P78,

6 quai Watier, 78401 Chatou, France

Received 6 January 2006; accepted 7 September 2006Editorial handling by D. Polya

Available online 21 November 2006

Abstract

The retention (or release) of a liquid compound on a solid controls the mobility of many substances in the environmentand has been quantified in terms of the ‘‘sorption isotherm’’. This paper does not review the different sorption mechanisms.It presents the physical bases underlying the definition of a sorption isotherm, different empirical or mechanistic models,and details several experimental methods to acquire a sorption isotherm. For appropriate measurements and interpreta-tions of isotherm data, this review emphasizes 4 main points: (i) the adsorption (or desorption) isotherm does not provideautomatically any information about the reactions involved in the sorption phenomenon. So, mechanistic interpretationsmust be carefully verified. (ii) Among studies, the range of reaction times is extremely wide and this can lead to misinter-pretations regarding the irreversibility of the reaction: a pseudo-hysteresis of the release compared with the retention isoften observed. The comparison between the mean characteristic time of the reaction and the mean residence time ofthe mobile phase in the natural system allows knowing if the studied retention/release phenomenon should be consideredas an instantaneous reversible, almost irreversible phenomenon, or if reaction kinetics must be taken into account. (iii)When the concentration of the retained substance is low enough, the composition of the bulk solution remains constantand a single-species isotherm is often sufficient, although it remains strongly dependent on the background medium. Athigher concentrations, sorption may be driven by the competition between several species that affect the composition ofthe bulk solution. (iv) The measurement method has a great influence. Particularly, the background ionic medium, thesolid/solution ratio and the use of flow-through or closed reactor are of major importance. The chosen method shouldbalance easy-to-use features and representativity of the studied natural conditions.� 2006 Elsevier Ltd. All rights reserved.

0883-2927/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.apgeochem.2006.09.010

* Corresponding author. Address: Atomic Energy Commission, Tracers Technology Laboratory, 38054 Grenoble Cedex, France. Fax:+33 4 38 78 51 34.

E-mail address: [email protected] (G. Limousin).

mailto:[email protected]

250 G. Limousin et al. / Applied Geochemistry 22 (2007) 249–275

Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2501. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2512. Reaction kinetics and thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

2.1. Relation between thermodynamics and kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2522.2. Kinetic hysteresis and pseudo-irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

3. Classification and modeling of the isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
3.1. The four main types of isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
3.1.1. The ‘‘C’’ isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2553.1.2. The ‘‘L’’ isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2553.1.3. The ‘‘H’’ isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.1.4. The ‘‘S’’ isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

3.2. Modeling of concave isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.2.1. The Freundlich models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.2.2. The Langmuir models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

3.3. Generalized modeling of any isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2573.4. Isotherms of uncharged organic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583.5. The ion exchange isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2593.6. Surface complexation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.7. How to choose among so many models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

4. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.1. Influence of the experimental conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.1.1. The solid/solution ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.1.2. Closed reactor versus open flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.1.3. The composition of the background solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

4.2. Description, advantages, and disadvantages of the methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.2.1 The batch method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.2.2. The flow-through methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

0. Introduction

Among all phenomena governing the mobility ofsubstances in aqueous porous media and aquaticenvironments, the transfer of substances from amobile phase (liquid or gaseous) to a solid phaseis a universal phenomenon. That is the reason whythe ‘‘isotherm’’, a curve describing the retention ofa substance on a solid at various concentrations, isa major tool to describe and predict the mobilityof this substance in the environment. These reten-tion/release phenomena are sometimes stronglykinetically controlled, so that time-dependence ofthe sorption isotherm must be specified.

This paper is limited to the retention/release ofliquid solutes on natural solid, mineral or organic,materials. The focus is on the interactions of min-

eral or organic compounds (trace elements, radio-nuclides, other mineral species such as plantnutrients, and organic compounds such as traceorganic non-aqueous or soluble contaminants) withnatural mineral or organic materials such as soil,aquifer or geologic material, and suspended parti-cles present in porous media, rivers, lakes andoceans. Many excellent books and papers havereviewed the literature on these mechanisms (e.g.Sposito, 1984; Sposito, 1989; McBride, 1994; Bolanet al., 1999; Pignatello, 2000; Huang et al., 2003), sothat this paper deals with chemical mechanisms onlywhen it is necessary for understanding their relationwith the isotherm. In contrast, few papers havereviewed some aspects of isotherms themselves(e.g. Schweich and Sardin, 1981; Hinz, 2001), whichjustifies gathering the theoretical and experimental

G. Limousin et al. / Applied Geochemistry 22 (2007) 249–275 251

knowledge on sorption isotherms applied togeochemistry.

This review provides some definitions, thermody-namics and kinetics of the reactions involved, thengives an overview on the empirical and mechanisticmodeling of both sorption isotherms and kinetics ofretention/release, and finally discusses variousexperimental methods currently available to mea-sure sorption isotherms and reaction kinetics.

1. Definitions

When the retention of a solute on solid particlesis investigated, the remaining solute concentrationof the compound C (mol L�1 or kg L�1) can becompared with the concentration of this compoundretained on solid particles Q (mol kg�1 or kg kg�1),as shown in Fig. 1. The relationship Q = f(C) isnamed the ‘‘sorption isotherm’’. The uniqueness ofthis relation requires several conditions to be met:(i) the various reaction equilibria of retention/

final state a

retention(adsorption)

unequilibrated initial state 1

reactive compound

liquid phase initial concentration: Ca0

liquid phase final concentration: C solid

final conc

solid phase initial concentration: Qa0

Fig. 1. Schematic view of the adsorption and desorption phenomena.calculated by difference between the initial liquid concentration (Ca0 orconcentration (Qa0 or Qb0) is negligible or previously measured.

release must have been reached, and (ii) all otherphysico-chemical parameters are constant. Theword ‘‘isotherm’’ was specifically chosen becauseof the influence of the temperature on sorption reac-tions; temperature must be kept constant and spec-ified (e.g. Cornelissen et al., 1997; Werth andReinhard, 1997).

Most of the time, the concentration of the com-pound retained on the solid is calculated by differ-ence between the initial solute concentration Ca0

and the final solute concentration C (Fig. 1). Inthe case of retention stage, the solid concentrationat equilibrium Q (mol kg�1) is given by Eq. (1)

Q ¼ VmðCa0 � CÞ þ Qa0 ð1Þ

with V being the volume of solution (L), m is the so-lid mass and Qa0 (mol kg�1) is the concentration ofthe compound initially retained by the solid, whichmust be measured or shown to be negligible. Inthe case of a release stage of a compound that is

t equilibrium

release(desorption)

unequilibrated initial state 2

phase: entration: Q

liquid phase initial concentration: Cb0

solid phase initial concentration: Qb0

This scheme shows that the final solid concentration Q can beCb0) and the final liquid concentration C only if the initial solid


initially present on the solid phase (Fig. 1), thequantity of this initially retained compound Qb0

must be measured previously. The solid concentra-tion at equilibrium can then be calculated by theabove equation, replacing Ca0 by Cb0 and Qa0 byQb0 (Fig. 1).

In 1888, this approach to represent solute reten-tion on solids was invented by Van Bemmelen(1888). This isotherm often cannot of itself provideinformation about the type of reaction involved(Sposito, 1984, p. 122; Veith and Sposito, 1977;Scheidegger and Sparks, 1996). For example, theretention can be either due to surface retention with-out creating three-dimensional structure or to pre-cipitation of a new solid phase (McBride, 1994, p.141). Sposito proposed to speak about ‘‘sorption’’to take into account any type or retention (Sposito,1984, p. 122). The precipitation itself is ofteninduced by surface adsorption (the surface actingas a ‘‘template’’) and thus called ‘‘surface enhancedprecipitation’’ (Wersin et al., 1994; Ford et al.,2002). This phenomenon was shown by James andHealy (1972) and is of major importance in the for-mation of new solid phases in soils and geologicmedia (Ford et al., 2002). Due to this diversity ofretention mechanisms, it could be more appropriateto use the term ‘‘retention isotherm’’ rather than‘‘adsorption isotherm’’. Some authors even proposethe term ‘‘curve of disappearance of the solute’’since the retention is often measured by differencebetween the initial and final solute concentration(Schweich and Sardin, 1981). It is important to keepin mind that the retention/release mechanisms arevarious and often intricate (Strawn and Sparks,1999). From now on, the terms ‘‘sorption’’,‘‘adsorption’’, and ‘‘desorption’’ will be used in their

molar free energy: G (J mol-1)

standard energy of the reaction: ΔG 0

activation energy of adsorption: Ea 1

Fig. 2. Energetic state of the solid/solute system during a reaction of

broadest sense (i.e. retention/release) unless other-wise indicated.

2. Reaction kinetics and thermodynamic equilibrium

2.1. Relation between thermodynamics and kinetics

A sorption isotherm is often assumed to be con-structed from data obtained at reaction equilibrium.In that case, the adsorption isotherm (made frommeasurements of progressive adsorption up to theequilibrium) should be the same as the desorptionisotherm (made from measurements of progressiverelease into the solution up to the equilibrium)because the theory of thermodynamic equilibriaassumes a complete reversibility of the chemicalreactions and so a unique (C,Q) pair (Strawn andSparks, 1999).

However, sorption mechanisms are driven byseveral various kinetically controlled reactions orphysical phenomena, which have (with redox reac-tions) the largest variability of reaction times: fromfew seconds to many years (Sparks, 2000). Fig. 2shows the energetic variations that occur duringadsorption or desorption.

In Fig. 2 is illustrated the standard free energy ofthe reaction DG0 (J mol�1), which is the differencebetween the initial state (free solute compound)and the final equilibrated state (adsorbed com-pound). Thus, DG0 is negative since the energy ofthe system decreases during the reaction to reach amore stable state. The relation between DG0 andthe reaction constant K (dimensionless) is given byEq. (2) (McBride, 1994, p. 14)

K ¼ e�DG0

RT ð2Þ

reaction pathway: ξ (mol)

activation energy of desorption: Ea -1

adsorption or desorption (adapted from McBride, 1994, p. 27).


with R being the gas constant (J mol�1 K�1) and T

the temperature (K).In Fig. 2, it is also shown that the thermody-

namic equilibrium is the result of a competitionbetween two contrary mechanisms, represented bythe activation energy of adsorption E1 and desorp-tion E�1. The Arrhenius law (Eq. (3)) provides therelation between the kinetic constants k (dimension-less) and the activation energy E (J mol�1)(McBride, 1994, p. 26)

k ¼ De�ERT ð3Þ

with D being a dimensionless collision probabilityfactor.

Fig. 3. Illustration of different sorption mechanisms of metal ions (Mcomplex on exchange sites located on basal planes (hydrated metal) ametal); or (b) inclusion of the metal into crystal structure by coprecipita1999b, 2001).

Hence, the relation between the reaction energyof the reaction and the activation energies (Eq. (4))

DG0 ¼ E1 � E�1 ð4Þcan be directly linked with the relation between thethermodynamic constant and the two kinetic con-stants (Eq. (5))

K ¼ k1

k�1

ð5Þ

By interpreting Fig. 2, it can be concluded that areaction can be thermodynamically spontaneous(DG0 negative, K > 1), whereas it can be (i) fast,either for adsorption or for desorption (low E1

e) on clay minerals by (a) adsorption by outer sphere surfacend as inner sphere surface complexes on the edges (dehydratedtion; or (c) precipitation of a new solid phase (after Schlegel et al.,


and low E�1) or (ii) slow for desorption (high E�1)or even (iii) slow for adsorption and for desorption(high E1, higher more E�1).

2.2. Kinetic hysteresis and pseudo-irreversibility

It is known that several metals and other heavyelements often show a fast kinetic adsorption reac-tion by outer-sphere ion exchange followed by slowadsorptions with specific interactions (Aston andDuursma, 1973; Amacher et al., 1986; Bangashet al., 1992; Schlegel et al., 1999b: see Fig. 3). Inthe case of slow kinetics for desorption (e.g. Nyffeleret al., 1984; Amacher et al., 1986; McLaren et al.,

Q

C

desorption isotherm

adsorption isotherm

Fig. 4. Pseudo-hysteresis of the desorption isotherm comparedwith the adsorption isotherm when equilibrium is not reachedduring the adsorption or desorption stage.

Tim

Sr

con

cen

trat

ion

(m

oL-1

)

0 200 400 600

0.010

0.008

0.006

0.004

0.002

0.000

Fig. 5. Breakthrough curve of Sr in an organic soil (from Limousin,erroneously to an irreversible adsorption for half of the Sr injected. Ho(while not present for the inert tracer) indicates that desorption is very

1986, 1998; Bruemmer et al., 1988; Bibak et al.,1995) or for adsorption, too (e.g. Kuo and Mikkel-sen, 1980; Padmanabham, 1983; Schultz et al., 1987;Backes et al., 1995; Lookman et al., 1995; Smith andComans, 1996; Schlegel et al., 1999a,b), it is some-times difficult to know if thermodynamic equilib-rium is reached (Smith and Comans, 1996; Sparks,1998). If not, the superimposition between adsorp-tion and desorption isotherms does not match(Fig. 4). This mismatch between adsorption anddesorption isotherms is called ‘‘pseudo-hysteresis’’or ‘‘kinetic hysteresis’’ (Strawn and Sparks, 1999).A more convenient term could be ‘‘apparent irre-versibility’’ (Van Bladel and Laudelout, 1967;McBride, 1994, p. 91; Ford et al., 2002). The com-mon term ‘‘irreversibility’’ can lead to the wrongconclusion that a part of the solute cannot be des-orbed at all, whereas it could be a matter of time(Van Bladel and Laudelout, 1967; Ogwada andSparks, 1986a,b,c; Comans, 1987; Young et al.,1987; Verburg and Baveye, 1994). On the otherhand, the saturation capacity of the solid duringadsorption can be underestimated if equilibrium isnot reached (Harter, 1984). The characteristic timeof desorption can be several orders of magnitudedifferent from the characteristic time of adsorption(Bangash et al., 1992; Ainsworth et al., 1994; Smithand Comans, 1996). The adsorbed compound canreact, in a second stage, with the solid by (i) slowdiffusion inside the solid (e.g. Boyd et al., 1947b;

[Sr]0 = 0.5 mol L-1

Sr mass balance after 1400 mn: 51%mean residence time of the inert tracer: 99 mn

[Sr]0 = 0.5 mol L-1


[Sr]0 = 0.5 mol L-1


[Sr]0 = 0.5 mol L-1


[Sr]0 = 0.5 mol L-1


[Sr]0 = 0.5 mol L-1


e (mn)

Srinert tracer (tritiated water)

800 1000 1200 1400

2006). If considering the mass balance only, one could concludewever, the presence of a thick tail on the Sr breakthrough curveslow but possibly total.


Bruemmer et al., 1988; Fuller et al., 1993; Coughlinand Stone, 1995; Burgos et al., 1996; Pignatello andXing, 1996; Axe and Anderson, 1997; Pignatello,2000), or (ii) inner-sphere surface complexation(see Fig. 3a; e.g. Benjamin and Leckie, 1981; Leh-mann and Harter, 1984; Grossl et al., 1994; Schlegelet al., 1999b; Manceau et al., 2000), or (iii) crystal-lization of new solid phases (see Fig. 3b and c; e.g.Chrisholm-Brause et al., 1990; Waychunas et al.,1993; Charlet and Manceau, 1994; Manceau et al.,1999; Schlegel et al., 2001). The duration of desorp-tion is increased by the time needed to reverse thesereactions. This highlights the importance of know-ing the history of a pollution and taking intoaccount the ‘‘ageing’’ effect (Strawn and Sparks,1999). For the description and the prediction ofdynamic phenomena (diffusion through clay materi-als, advective–dispersive transport through soils andaquifers, particle transport in natural water . . .), themeasurement of the mean residence time of water inthe system is required in order to compare it withthe mean characteristic time of the reaction. Forexample, an apparent ‘‘irreversible’’ adsorption ofa long-life radionuclide can often be consideredeffectively irreversible in aquifers because the watervelocity is high, whereas the same adsorption reac-tion should be considered completely reversible fora long-term waste disposal. Such slow desorptionphenomena can be detected by a thick tail of thebreakthrough curve during a flow-through experi-ment, which is on the contrary not present for aninert tracer, as shown in Fig. 5 (Limousin, 2006).In such a case, it is important to lower the back-ground noise as far as possible.

(a) The “C” isotherm (b

(c) The “H” isotherm (d

Q

C

Q

C

Fig. 6. The four main types of isoth

3. Classification and modeling of the isotherms

3.1. The four main types of isotherms

Giles et al. (1974) proposed a general modellingof sorption isotherms, in which 4 particular casesare now used as the 4 main shapes of isotherm com-monly observed (Fig. 6).

3.1.1. The ‘‘C’’ isotherm

The curve is a line of zero-origin (Fig. 6a). Itmeans that the ratio between the concentration ofthe compound remaining in solution and adsorbedon the solid is the same at any concentration. Thisratio is usually named ‘‘distribution coefficient’’ or‘‘partition coefficient’’: Kd or Kp (L kg�1). The‘‘C’’ isotherm is often used as an easy-to-useapproximation (for a narrow range of concentrationor very low concentrations such as observed fortrace pollutants) rather than an accurate descrip-tion. But the simplicity of this isotherm must notjustify its use without verification, otherwise it couldlead to erroneous conclusions. For example, if thesolid has a limited quantity of adsorption sites, theisotherm could be nonlinear because of a possiblesaturation plateau.

3.1.2. The ‘‘L’’ isothermThe ratio between the concentration of the com-

pound remaining in solution and adsorbed on thesolid decreases when the solute concentrationincreases, providing a concave curve (Fig. 6b). Itsuggests a progressive saturation of the solid. Oneusually makes two sub-groups: (i) the curve reaches

) The “L” isotherm

) The “S” isotherm

Qwith strict plateau

without strict plateau

C

point of inflection

Q

C

erms (after Giles et al., 1974).


a strict asymptotic plateau (the solid has a limitedsorption capacity), and (ii) the curve does not reachany plateau (the solid does not show clearly a lim-ited sorption capacity). But it often appears practi-cally difficult to know if an isotherm belongs tothe first or to the second sub-group.

3.1.3. The ‘‘H’’ isotherm

This is only a particular case of the ‘‘L’’ isotherm,where the initial slope is very high (Fig. 6c). Thiscase was distinguished from the others because thecompound exhibits sometimes such a high affinityfor the solid that the initial slope cannot be distin-guished from infinity, even if it does not make sensefrom a thermodynamic point of view (Toth, 1995).

3.1.4. The ‘‘S’’ isotherm

The curve is sigmoidal and thus has got a point ofinflection (Fig. 6d). This type of isotherm is alwaysthe result of at least two opposite mechanisms.Non-polar organic compounds are a typical case:they have a low affinity with clays. But as soon asa clay surface is covered by these compounds, otherorganic molecules are adsorbed more easily (e.g.Karimi-Lotfabad et al., 1996; Pignatello, 2000). Thisphenomenon is called ‘‘cooperative adsorption’’(Hinz, 2001) and is also observed for surfactants(e.g. Smith et al., 1990; Smith and Galan, 1995; Gro-isman et al., 2004). The presence of a soluble ligandcan also provide a sigmoidal isotherm for metallicspecies. At low metal concentrations, the adsorptionis limited by the presence of the ligand. The ligandmust be saturated and then the adsorption occursnormally (Sposito, 1984, p. 116). The point of inflec-tion illustrates the concentration for which theadsorption overcomes the complexation.

3.2. Modeling of concave isotherms

3.2.1. The Freundlich models

3.2.1.1. Simple Freundlich model. The concave iso-therm (‘‘L’’ or ‘‘H’’ isotherms) is the most widelymet isotherm. The first model is empirical (VanBemmelen, 1888; Freundlich, 1909) and is basedon the following relation between the adsorbedquantity Q and the remained solute concentrationC (Eq. (6))

Q ¼ FCn ð6Þwith F (L kg�1) and n (dimensionless) being twoconstants (n < 1). This equation is easily linearizable(Eq. (7))

log Q ¼ log F þ n log C ð7ÞA graph with logC as x-axis versus logQ as y-axisprovides a line of slope n and intercepts the y-axisat logF. According to the Freundlichequation, the isotherm does not reach a plateau asC increases.

3.2.1.2. Modified Freundlich model for competitive

adsorption. It is well known that adsorption is sub-jected to competition between several species (e.g.Murali and Aylmore, 1983c; Roy et al., 1986; Ban-gash et al., 1992). In order to take into account com-petitive phenomena, numerous modified Freundlichmodels have been built, often empirical without anyphysical basis. For example, several modified Fre-undlich isotherms (Fritz and Schundler, 1981;Sheindorf et al., 1981) generalize the Freundlichequation to m competitive species (Eq. (8))

Qi ¼ F iCi

Xm

j¼1

ai;jCj

!ni�1

ð8Þ

where ai,j is the dimensionless competition coeffi-cient of species i in the presence of species j, andFi and ni are the coefficients of the Freundlich iso-therm of the species i. This formula has been suc-cessfully applied to the adsorption of cations(Guttierrez and Fuentes, 1991) and anions in soils(Roy et al., 1986) and to the adsorption of organiccompounds on activated organic carbon (Sheindorfet al., 1982).

3.2.2. The Langmuir models

3.2.2.1. Simple Langmuir model. Another very com-mon model is based on reaction hypotheses (Lang-muir, 1918). The solid is assumed to have alimited adsorption capacity Qmax. All the adsorp-tion sites (i) are assumed to be identical, (ii) each siteretains one molecule of the given compound and(iii) all sites are energetically and sterically indepen-dent of the adsorbed quantity. Then, the followingreaction is considered:

free siteþ solute $ surface complex

Since activities of adsorbed species are notclearly defined thermodynamically, the mass actionlaw cannot be directly applied to this reaction.Nevertheless, it has been proposed to assume thesurface activity coefficients equal to unity and tocalculate the activities with conditional stabilityconstant, where Q is the solid concentration of


the retained compound on the solid and Qmax � Q

is the solid concentration of the free adsorptive site(Eq. (9))

L ¼ ½surface complex�½solute�½free site� ¼

QCðQmax � QÞ ð9Þ

Therefore, the ‘‘Langmuir’’ isotherm is (Eq. (10))

Q ¼ Qmax

LC1þ LC

ð10Þ

It can be linearized by (Eq. (11))

QC¼ QmaxL� LQ ð11Þ

A graph with Q as x-axis and Q/C as y-axis pro-vides a line of slope �L and intercepts the y-axis atQmaxL. According to the initial assumptions, theisotherm reaches a plateau Qmax (contrary to theFreundlich isotherm). The constant QmaxL is the ini-tial slope of the isotherm and QmaxL is often used asa distribution coefficient (Kd) when the concentra-tions are low enough to justify this approximation.The constant L corresponds to the affinity of thecompound for the solid, while Qmax corresponds tothe adsorption capacity of the solid.

3.2.2.2. Modified Langmuir models for multisite or

competitive adsorption. Although the Langmuir iso-therm is widely used (Travis and Etnier, 1981), itslinearization sometimes gives poor results: the graphQ/C versus Q often shows a convex curve. Severalexplanations have been suggested. The first one isthe existence of several types of adsorption sites.In that case, it is possible to generalize the Langmuirmodel to p couples (Li,Qmax,i), each of them corre-sponding to one type of site (Eq. (12))

Q ¼Xp

i¼1

Qmax;i

LiC1þ LiC

ð12Þ

This equation can be linearized part by part by usingEq. (11), providing each couple (Li,Qmax,i). Theadsorption of phosphate on soils was almost per-fectly described by Eq. (12) with two couples(L1,Qmax,1) and (L2,Qmax,2) (Holford et al., 1974).But several authors have demonstrated that the reci-procity is not true: a perfect adjustment to this modelcannot demonstrate a priori the existence of severaltypes of sites (Posner and Bowden, 1980; Sposito,1982) even if other experiments showed that phos-phate is effectively adsorbed by two types of sites.The second explanation to the poor linearity of thefunction Q/C = f(Q) is the decrease of adsorption en-

ergy as the fraction of occupied sites increases. In thatcase, the independence between two sites is not valid.The third explanation proposed is the competitionbetween two species. Some continuous open-flowtechniques allow flushing the competitive species assoon as they are displaced, limiting their influence(see Section 4.1.2). But it is possible to model an ex-change between the added species i and the displacedspecies j (which remains in competition for adsorp-tion). Following this hypothesis, the modification ofthe Langmuir isotherm yields Eq. (13) (Boyd et al.,1947a)

Qi ¼ Qmax;i

LiCi

1þ LiCi þ LjCjð13Þ

This equation was then generalized (Murali andAylmore, 1983a,b,c) for the competition between q

species (Eq. (14))

Qi ¼ Qmax;i

LiCi

1þPq

j¼1LjCjð14Þ

According to this formula, the saturation capacityQmax is not affected by competition (the differentspecies being adsorbed by the same adsorptive sites)and a simple Langmuir isotherm can be used to ob-tain it. On the contrary, the affinity constant Li isinfluenced by the competitive species (Harter andBaker, 1977). Murali and Aylmore (1983a) used dif-ferent particular cases to simplify this equation.However, when the competitive phenomenon ision exchange, the ion exchange isotherm approachshould be preferred (see Section 3.5).

3.3. Generalized modeling of any isotherm

In most cases where the concentration of thestudied compound is higher than a trace concentra-tion, neither the Langmuir nor the Freundlich iso-therms are consistent with the data and morecomplicated models must be applied (Kinniburgh,1986). Hinz (2001) proposed an equation that coulddescribe any type of isotherm (Eq. (15))

Q ¼ Qmax

Xx

i¼1

fi

Ysi

j¼1

Ai;jCpi;j

1þ Bi;jCqi;j

� �ri;j

ð15Þ

where Qmax denotes the asymptotic amount ofadsorption at high concentrations, fi is the fractionof sites of type i (whereas the total number of differ-ent types of site is x), and si gives the number ofinteraction terms between the different types of sites.Ai,j and Bi,j are empirical affinity constants and pi,j,qi,j and ri,j are dimensionless empirical parameters.


Although this equation is fully empirical andincludes many fitting parameters, it has the advan-tage of decomposing any isotherm into differenttypes of sites.

On the basis of the Langmuir model, anothergeneralized model for any isotherm was constructed(Sips, 1948, 1950; Sposito, 1984, p. 120; Nederlofet al., 1990). The basic idea is to consider any iso-therm as an integral of Langmuir isotherms(Fig. 7a). This integral has a density function g(L)which corresponds to the statistical distribution ofthe affinity constant L (Eq. (16))

Q ¼Z þ1

�1gðLÞ LC

1þ LCdK ð16Þ

with g being a miscellaneous density function.Conceptually, each adsorption site provides an

elementary isotherm having its own affinity L andcapacity g(L). The complete isotherm is seen asthe sum of all the elementary isotherms, as shownin Fig. 7a. The function g has been named a‘‘weighting function’’ (Sposito, 1980), a ‘‘site affinitydistribution function’’ (Kinniburgh et al., 1983), ora ‘‘frequency distribution of the local affinity coeffi-cient L’’ (Perdue and Lytle, 1983).

By choosing the accurate density function g, anytype of isotherm can be described by Eq. (16) (Sposito,1984, p. 120; Hinz et al., 1994). In the case of the simpleLangmuir isotherm, one can isolate Qmax (Eq. (17))

Qmax ¼Z þ1

�1gðLÞdL ð17Þ

According to Eq. (16), g requires to be a Dirac’sfunction (Eq. (18)) in order for Qmax to be aconstant

logQ

C

complete isotherm (sum of all the Langmuir isotherms)

elementaryLangmuirisotherms

(a) isotherm decomposition (general case)

Fig. 7. Conceptual decomposition of an isotherm into several elementarsaturation capacity). The Freundlich isotherm is a particular case where1984, p. 120).

gðlnðLÞÞ ¼ QmaxdðL� LmÞ ð18Þwith d being the Dirac’s function and Lm the mostprobable L.

The Langmuir isotherm is consequently the sumof isotherms having the same constant Lm. If theFreundlich isotherm is needed, g must be a functionwhose curve closely resembles a log-normal distri-bution (not detailed here). Hence, one can concep-tualize the Freundlich isotherm as the sum ofindividual Langmuir isotherms (Fig. 7b) with alog-normal distribution of their affinity constants(Sposito, 1984, p. 121; Kinniburgh et al., 1983; Hinzet al., 1994). The same result would be obtained byassuming that the Freundlich isotherm is a derivedLangmuir isotherm where the adsorption energydecreases logarithmically as the fraction of occupiedsites Q/Qmax increases (Hasley and Taylor, 1947).

3.4. Isotherms of uncharged organic compounds

Excellent reviews about the adsorption anddesorption mechanisms of organic compounds havebeen written (Pignatello, 2000; Huang et al., 2003).Like many other adsorbed substances, the linearapproximation of the isotherm is often suitable atlow concentration (e.g. Schwartzenbach and Wes-tall, 1981; Groisman et al., 2004), so that a constantdistribution coefficient can be applied. Moreover,non-polar or at least uncharged organic compoundsare hydrophobic and mainly adsorb via Van derWaals attractions onto the hydrophobic part of solidorganic matter. So, their affinity for the solid is oftenexplained by their affinity for solid organic matteronly (e.g. Karickhoff et al., 1979; Schwartzenbach

Q

log C

Freundlich isotherm (sum of all the Langmuir isotherms)

elementaryLangmuirisotherms

(b) decomposition of a Freundlich isotherm (log-log scale)

slope <1

slopes = 1

y Langmuir isotherms (each of them having a distinct affinity andthe affinity constants are log-normally distributed (after Sposito,


and Westall, 1981). Although this statement is nowdiscussed (e.g. Luthy et al., 1997; Sheng et al.,2001), sorption of uncharged organic compoundson solid organic matter predominates in most cases,so that the distribution coefficient Kd is proportionalto the mass fraction of organic carbon in the solid foc

(kg kg�1) and to the affinity constant between thecompound and the organic carbon Koc (L kg�1)(Chiou et al., 1979), as described by Eq. (19)

Kd ¼ focKoc ð19ÞMoreover, Koc is often well correlated to any parti-tion coefficient of the compound between water andanother hydrophobic organic substance (or even itsconstant of solubility) by a logarithmic relationship(Eq. (20)). Most of the time, the chosen coefficient isthe easily available octanol/water partition coeffi-cient Kow (L kg�1)

logðKocÞ ¼ a logðKowÞ þ b ð20Þwith a and b being two constants. This relation wasverified for a wide variety of solids and unchargedorganic compounds (e.g. Karickhoff, 1981; Schwart-zenbach and Westall, 1981; Gerstl, 1990), or evenfor charged organic compounds for which thenon-polar part of the molecule is big enough.Although the parameters a and b are obviouslydependent on the nature of the solid organic matterand on the class of adsorbed organic compound(Rutherford et al., 1992), these parameters were of-ten found to remain constant for a wide range ofsoils and organic compounds. Thus, it becomes pos-sible to give an estimation of the adsorption of anyuncharged organic compound at low concentrationon any natural material with two easily measuredparameters: the organic carbon content of the mate-rial and the octanol/water partition coefficient ofthe compound.

Regarding the charged organic compounds thatdo not behave as described above, as well as regard-ing the behaviour of uncharged organic compoundsabove the limit of linearity of their isotherm, a list ofisotherms of several organic compounds in soils hasbeen provided (Weber and Miller, 1989).

3.5. The ion exchange isotherms

The mineral species (which are almost all chargedin aqueous solution) are mainly attracted by the sur-face charges of the solid. Thus, their adsorption isdriven by competing ion exchange phenomena.When the concentration of the studied ion is low

compared to the concentration of other competingions, the composition of the bulk solution can beconsidered constant and a single-species isothermcan be applied, although remaining strongly depen-dent on the composition of the bulk solution. How-ever, when the concentration of the studied ionreaches the same order of magnitude as other com-peting ions, the composition of the bulk solutioncan no longer be considered constant and a multi-species isotherm is needed.

It is possible to take into account the competitionbetween several ions with modified multi-speciesisotherms (see Sections 3.2.1.2 and 3.2.2.2). Muraliand Aylmore (1983b) simulated the consequencesof different competition features on the shape ofmulti-species equilibrium or kinetic Freundlichand Langmuir isotherms. The ion exchange iso-therm is another way to describe the competitionbetween two or more ions when their range of con-centration is wide. This specific isotherm does notdescribe Q versus C, but the molar fraction ofcharges eEk of the ion k adsorbed on the solid((molc kg�1)/(molc kg�1)) versus the molar fractionof charges Ek of the ion k remaining in solution((molc L�1)/(molc L�1)). The molar fraction ofcharges Ek of an ion k is defined by Eq. (21)

Ek ¼Nk

N total

ð21Þ

with Nk being the normality of ion k (molc L�1) andNtotal is the total normality (molc L�1). The normal-ity of the ion k is defined as its aqueous concentra-tion of charges (Eq. (22))

Nk ¼ mkCk ð22Þ

with mk being the valence of ion k (dimensionless),and the total normality being the sum of normalitiesof all present ions or, said differently, the total con-centration of charges in solution (Eq. (23))

N total ¼X

k

Nk ð23Þ

So, Ek and eEk vary between 0 and 1, according toEq. (21).

It is also assumed that the number of adsorptionsites (surface charges of the solid) is constant, and iscalled the ‘‘intrinsic charge’’ of the solid rint

(molc kg�1). For a cation (resp. anion) exchange,the adsorption capacity depends on the cationexchange capacity (resp. anion exchange capacity).But it is important to keep in mind that the cationor anion exchange capacity is an operational


parameter and not an intrinsic property of the solid.So, it cannot be defined as rint, which is a concep-tual parameter. The anion or cation exchangecapacity is strongly dependent on the measurementconditions such as pH or ionic strength (Charlet andSchlegel, 1999; Limousin and Tessier, 2003). To givea realistic value to rint, the best method is probablyto measure the anion or cation exchange capacitywith one of the studied ions itself.

The ion exchange isotherm is constructed froman exchange between two ions i and j, with Ntotal

remaining constant, otherwise the uniqueness ofthe isotherm is not valid (e.g. Singhal et al., 1978;Schweich and Sardin, 1981). Fig. 8 shows NH4/Caexchange isotherms on two minerals. In this exam-ple, the affinity of hydroxy-Al interlayered vermicu-lite is stronger for NH4 because the curve is locatedabove the non-preference curve (which representsthe equal affinity of the solid for NH4 and Ca),whereas vermiculite does not exhibit significantpreference for NH4 or Ca. For a homoionicexchange (two ions with same valence), the non-preference curve is simply the 1:1 line. However,the non-preference curve of an exchange betweentwo ions with different valences is not a line becausethe purely electrostatic attraction of an ion (withoutany specific affinity) is proportional to the exponen-tial of its valence, so that the ion of highest valenceis more adsorbed. Here the exchange between two

Fig. 8. Ion exchange isotherm: exchange between two heterova-lent ions NHþ4 and Ca2+ on vermiculite and hydroxy-aluminuminterlayered vermiculite, compared with the non-preference curveof a monovalent/divalent ion exchange. The total normality wasmaintained constant at 1 molc L�1 (from Evangelou and Lum-banraja, 2002).

ions i (with valence ai) and j (with valence aj) isdescribed

aj solid-iþ ai free j$ ai solid-jþ aj free i

By similarity with the law of mass action, the selec-tivity coefficient Kex is (Eq. (24))

Kex ¼½solid-j�ai ½free i�aj

½solid-i�aj ½free j�ai¼eEai

j EajieEaj

i Eaij

ð24Þ

Thus, a non-preference of the solid means thatKex = 1, which leads to the same fixed/free ratioðeEk=EkÞ for both ions only if ai = aj. However, Eq.(24) is a way to find a mathematical similarity withthe law of mass action, but it cannot be consideredas a thermodynamic formula: most of the time, ionexchange deviates from the ideal exchange behav-iour, so that the selectivity coefficient Kex varies sig-nificantly with Nk (e.g. Singhal et al., 1976). Thus,Kex values should always be provided with the rangeof Nk and ionic strength used for the measurement.Among many formulas modelling ion exchange, asimple analytical model was proposed to describeboth ion exchange with possible non-ideal behaviorand inner-sphere complexation (Fletcher and Spos-ito, 1989).

Sometimes the ion exchange isotherm has a pointof inflection. It means that the preferential affinityof the solid for one of the two ions shifts to the otherion when the fraction of charges occupied by thefirst ion reaches a certain value. It can be causedby the existence of two types of sites. A well knownexample is the K (or NH4)/Ca exchange on the 2:1

Fig. 9. K/Ca exchange isotherm on an Australian soil containing2:1 clay minerals. The total normality was maintained constant at0.2 molc L�1 (from Bloom and Mansell, 2001).


clays (Fig. 9). For a high fraction of Ca, the ditrigo-nal sites that are available to ion exchange have aspecific affinity for K, and so specifically adsorb thismonovalent cation. Then, for a high fraction of K(when all the available ditrigonal sites are occupiedby K), the other sites do not have specific affinityand the isotherm tends to reach the non-preferencecurve of a divalent/monovalent ion exchange (e.g.Beckett, 1964; Jardine and Sparks, 1984; Bloomand Mansell, 2001).

3.6. Surface complexation models

Although this paper is not aimed at reviewingsorption mechanisms, mechanistic models havebeen markedly improved during the last 20 yearsand are often used to construct sorption isothermsand to gain mechanistic interpretation of them,including the influence of ionic background (e.g.Sahai and Sverjensky, 1997; Criscenti and Sverjen-sky, 2002; Sverjensky, 2005). These surface com-plexation models are based on inner- or outer-sphere complexations, and can be distinguishedfrom each other by the way they represent the distri-bution of the electric potential around the chargedsurfaces. The main historical models are: the tri-ple-layer model (Davis et al., 1978), the constant

Table 1Some classical equilibrium models of adsorption with analytical formu

Curve shape Model

‘‘C’’ curve LinearLinear for the species i, in competition

‘‘L’’ or ‘‘H’’ curve FreundlichFreundlich with competition between tw

Freundlich with competition between m

Temkin

Rothmund–Kornfeld (exchange betwee

Langmuir

Langmuir with competition between tw

Langmuir with competition between q

Langmuir-Freundlich

Generalized Langmuir

Redlich–Peterson

Toth

Hinz

‘‘S’’ curve Sigmoidal Langmuir

C: aqueous concentration, Q: solid concentration.From Murali and Aylmore (1983a,b,c), Selim (1999) and Hinz (2001).

capacitance model (Stumm et al., 1980), the Sternvariable surface charge model (Bowden et al.,1980), the one-pK model (Van Riemsdijk et al.,1986), the generalized two-layer model (Dzombakand Morel, 1990), and the charge distribution model(Hiemstra and Van Riemsdijk, 1996). Good reviewshave been written on the physical bases and experi-mental applications of surface complexation models(e.g. Goldberg, 1992, 2004). Thanks to the increas-ing power of computers, it has become possible toapply these models to many codes such as thesefree-download programs: MinteqA2 (Allison et al.,1991), PhreeqC (Parkhurst and Appelo, 1999) orChess (Van der Lee and De Windt, 1999), whichare continuously improved and appear now to beessential tools for geochemistry.

3.7. How to choose among so many models?

Table 1 shows the diversity of the formulas thatexist to model sorption isotherms at equilibriumand Table 2 provides other formulas modelling thekinetic dependence of adsorption/desorption. Theseformulas can be included in solute transfer equa-tions such as through-diffusion or advection-disper-sion equations (Selim, 1992). The choice of theadsorption/desorption model can have a great influ-

lae

Formula

Q = KdC

with other species Qi ¼ KdCiPn

j¼1ai;jCj

Q = FCn

o species i and j Qi ¼ F iCiðai;jCjÞni�1

species Qi ¼ F iCiPm

j¼1ai;jCj

� �ni�1

Q = K1ln(C) + K2

n two ions i and j) QiQj¼ Kex

CiCj

� �n

QQmax¼ LC

1þLC

o species i and j Qi ¼ Qmax;iLiCi

1þLiCiþLjCj

species Qi ¼ Qmax;iLiCi

1þPq

j¼1LjCj

QQmax¼ LCn

1þðLCÞnQ

Qmax¼ LC

1þLC

� �n

QQmax¼ LC

1þðLCÞnQ

Qmax¼ LC

½1þðLCÞn �1n

Q ¼ Qmax

Pxi¼1fi

Qsij¼1

Ai;jCpi;j

1þBi;jCqi;j

� �ri;j

QQmax¼ LC

1þLCþSC

Table 2Some classical kinetic models of adsorption/desorption withanalytical formulae

Model Formula

First order dQdt ¼ h

q k1C � k�1Q

nth order dQdt ¼ h

q k1Cn � k�1Q

Langmuir kinetic dQdt ¼ h

q k1CðQmax � QÞ � k�1Q

Langmuir kinetic withcompetitionbetween n species

dQdt ¼ h

q k1;iCi Qmax �Pn

j¼1Qj

� �� k�1;iQi

Elovich dQdt ¼ k expð�PQÞ

Power dQdt ¼ h

q kCnQm

Mass transfer (fractionof immobile water)

dQdt ¼ h

q kðC � CimÞ

First order with partlypseudo-irreversibleadsorption

dQdt ¼ h

q kðC � CirrÞ

C: aqueous concentration, Q: solid concentration, t: time, h:volumetric water content, q: bulk density.From: Murali and Aylmore, 1983a,b,c; Selim, 1999.


ence on the prediction of solute transfer (e.g. Hinzet al., 1994; Brusseau, 1998). An ideal model verifies4 properties: it must be effective, comprehensive,realistic and predictive (Barrow and Bowden,1987). Fitting simultaneously all the parameters ofa model is obviously the worst solution (Westalland Hohl, 1980; Brusseau, 1998) because many dif-ferent mechanisms could provide the same break-through curve (see Section 4.2.2). A step-by-stepmethod must be applied, trying first to adjust thesimplest models (or the most adequate model, basedon a mechanistic knowledge of the system) to thedata and then choosing a more complex model ifneeded. If a new parameter is introduced into themodel without any physical justification, its accu-racy should be verified by testing its sensitivity indifferent conditions and its physical meaning mustbe concluded carefully. When it is not possible tofit each parameter separately, it is impossible to giveany physical meaning to the parameters, and anadequate criterion must be selected to discriminateeach model. Saiers and Hornberger (1996) proposedapplying an index named ‘‘model selection crite-rion’’ (Koeppenkastrop and Decarlo, 1993), orMSC (Eq. (25))

MSC ¼ ln

Pdi¼1wiðCi � CÞ2Pd

i¼1wiðCi � CmiÞ2

!� 2k

dð25Þ

where d is the number of data points, k is the num-ber of fitted parameters, Ci is the ith observed con-centration from the replicate experiments, C is the

average of observed concentrations, Cmi is the mod-el-calculated concentration corresponding to the ithreplication, and wi is a weighting factor (if needed).The model retained must have provided the highestMSC. This type of quantitative criterion has theadvantage of taking into account the number of fit-ted parameters (which should be as low as possible),as well as a classical least-square calculation.

4. Experimental methods

4.1. Influence of the experimental conditions

A sorption isotherm does not have any intrinsicthermodynamic definition: its significance dependson the conditions from which it is obtained. So,the measurement method has a strong influence onthe results and a brief description of it shouldalways be provided with the data (Schweich andSardin, 1981).

4.1.1. The solid/solution ratio

In many cases, the ratio of solid mass versus solu-tion volume should theoretically not influence theproportion of adsorbed compound. And most ofthe time, this parameter does not effectively influ-ence the shape of the isotherm if it remains in thesame order of magnitude. However, numerousauthors observed a significant and nonlinear depen-dence of the solid/solution ratio on the amount ofadsorption (Aston and Duursma, 1973; O’Connorand Connolly, 1980; Di Toro and Horzempa,1982; Voice et al., 1983; Di Toro, 1985; Tan andTeo, 1987; Boesten and Van der Pas, 1988; Bangashet al., 1992; Bajracharya et al., 1996; Porro et al.,2000). Despite a few contradictory studies (Kossand Kim, 1990), adsorption is often observed todecrease with the solid concentration. This phenom-enon is called the ‘‘solid effect’’. The main reasonsproposed are (i) the occupied volume of the sus-pended particles (Celorie et al., 1989) and (ii) theiraggregation (Voice et al., 1983; Di Toro et al.,1986) that would prevent an optimal adsorption ofthe solutes. For the particular case of clay materials,the cationic exclusion volume increases as compac-tion increases, so that the solid/solution ratio hasan additional influence. The best experimentalchoice is a solid/solution ratio that is representativeof the natural conditions. In the case of water withsuspended particles, the solid concentration is oftentwo low to be used in batch. In contrast, the solid/solution ratio of soils and other geologic media is


too high to be used in batch (e.g. the solid/solutionof soils is typically 1:1, whereas the solid/solutionratio of aquifers is closer to 3:1 and the packed claybarrier materials have a solid/solution ratio higherthan 10:1, up to the limit where only adsorbed wateris present), but can be achieved in column experi-ments. For batch experiments with these naturalporous media, the range 1 g of solid for 2 mL to1 g of solid for 4 mL is advised (Porro et al.,2000). But for strongly adsorbed compounds, thisideal solid/solution ratio is sometimes too high todetect the compound remaining in solution. In con-clusion, the choice of an adequate solid/solutionratio consists of finding a good intermediatebetween experimental constraints and representativeconditions.

4.1.2. Closed reactor versus open flow

The isotherms obtained with flow-through exper-iments are often consistent with batch data (e.g.Bond and Phillips, 1990; MacIntyre et al., 1991).However, the removal (or not) of the solution, i.e.maintaining (or not) a constant composition of thesolution, can influence the quantity of adsorbedcompounds. In batch or in re-injected flow methods,the displaced competitive substances remain in solu-tion, and thus may interact with the solid. In con-trast, the open-flow methods supply the systemwith a constant solution and the displaced sub-stances are flushed out and thus do not competefor adsorption. Therefore, adsorption should be

0.01

0.10

1.00

10.00

0.01 0.10 1.

C (µm

Q (

µmo

l g-1

)

Fig. 10. Comparison of two uranyl isotherms obtained by the batch or1998). The authors show that the lower adsorption in batch is due to thein column because of continuous flushing.

higher for the open-flow methods such as columnor stirred-flow through reactor (Akratanakulet al., 1983). Some authors have observed this phe-nomenon (Akratanakul et al., 1983; Miller et al.,1989b; Porro et al., 2000; Gabriel et al., 1998: seeFig. 10). But many other studies show that theadsorption is higher in batch than in open-flow col-umn (Boekold and Van der Zee, 1992; Grolimundet al., 1995; Bajracharya et al., 1996). The reasonsfor this paradoxical behaviour are mainly (i) thepresence of immobile water in the column, whichacts as a kinetic barrier (Nkedi-Kizza et al., 1985;Maraqa et al., 1997; Plassard et al., 2000), (ii) thedifference of solid/solution ratio between batchand column (O’Connor and Connolly, 1980; Voiceet al., 1983; Di Toro, 1985), or (iii) the unachieve-ment of chemical equilibrium in column if the meanresidence time is significantly lower than the meanreaction time (Bilkert and Rao, 1985; Brusseauet al., 1991; Maraqa et al., 1998; Thomsen et al.,1999; Altfelder et al., 2001). The spatial chemicalheterogeneities of the medium have also beeninvoked (Chrysikopoulos et al., 1990; Brusseauand Zachara, 1993; Wise, 1993; Fesch et al., 1998),but they can be neglected compared with the heter-ogeneity of the pore water velocities (Brusseau andZachara, 1993; Johnson et al., 2003). Maraqa(2001a) investigated the adsorption of an organiccompound on a sandy material in batch, open-flowcolumn and recirculated-flow column. The distribu-tion coefficients obtained in batch or with recircu-

00 10.00 100.00

column

batch

= standard error

ol L-1)

column method in a silica–goethite medium (after Gabriel et al.,competition between uranyl and dissolved silica, while impossible

-9

-8

-7

-6

-5

-4

-10 -9 -8 -7 -6 -5 -4 -3

flow-through, non-equilibrated water (Martin-Garin et al., 2003)

flow-through, pre-equilibrated water (Martin-Garin et al., 2003)

batch (McBride, 1980)

batch (Davis et al., 1987)

batch (Papadopoulos & Rowell, 1988)

batch (Zachara et al., 1991)

batch (Van der Weijden, 1995)

1

1

log C (mol L-1)

log

Q (

mo

l kg

-1)

Fig. 11. Effect of the background composition of the solution on the adsorption isotherm of Cd onto calcite. This effect was either tested inbatch (with different background compositions) or in stirred flow-through reactor with pre-equilibrated or unequilibrated water. Eachconcentration of the isotherms constructed from stirred-flow through experiments corresponds to one complete breakthrough with thesame injected concentration. The solid concentration is given in mol m�2 to take into account the differences in specific surface area of thesamples (after Martin-Garin et al., 2003). See above-mentioned references for further information.


lated-flow column are similar whereas they are 30–170% higher in open-flow column. Therefore, thedifferences usually observed between batch andflow-through methods could not be due to the differ-ent solid/solution ratios but only to the removal ofthe solution. Since many systems of interest areopen and dynamic in nature (solute diffusion inclays, solute transfers in soils and aquifers, particletransport in lakes or rivers. . .), one can think thatopen-flow methods are closer to these natural condi-tions and thus should be preferred as often as possi-ble (Schweich et al., 1983; Sparks, 1985), taking intoaccount the background composition of the injectedsolution.

4.1.3. The composition of the background solutionAs indicated in Section 3.5, the ionic and molec-

ular background of the solution is of major impor-tance, especially when adsorption or desorption isgoverned by competing mechanisms such as ionexchange. If the solution is well equilibrated withthe solid, the differences between experiments per-formed with deionised or pre-equilibrated watercan be negligible. This condition can be completedfor batch experiments with a solid/solution ratiohigh enough. However, for any other condition(batch experiment with low solid/solution ratio orflow-through experiment), the background compo-sition of the solution must be as close as possibleto what is expected or measured in natural condi-tions. For example, Martin-Garin et al. (2003),showed the influence of the pre-equilibration on

the adsorption of Cd on calcite in stirred-flowthrough experiments (Fig. 11) and managed todescribe the observed differences among their resultsand other batch data by modelling the effects of pHand Ca concentration.

4.2. Description, advantages, and disadvantages of

the methods

Table 3 summarizes the main characteristics,advantages and disadvantages of each method foran appropriate purpose.

4.2.1. The batch method

The batch is far easier to use than any othermethod. The solid is shaken in the solution untilthe adsorption or desorption equilibrium is reached.Then, the remaining solute concentration ismeasured.

The disadvantages are numerous. The solid/solu-tion ratio is often either too high compared with thenatural conditions in rivers, lakes or seas, or too lowcompared with the natural conditions of porousmedia. Moreover, the hydrodynamic conditions ofnatural porous media are not met. A long termexperiment with continuous shaking can lead to sidereactions (Sposito, 1984, p. 113) such as the destruc-tion of particles, which prevents the study of veryslow reactions. As a consequence, the batch methodis very useful as a preliminary experiment butextrapolation to porous media requires otherinvestigations.

Table 3Main characteristics of experimental methods for the measurement of adsorption/desorption isotherms and kinetics

Method Solid/solutionratio

Adsorptionmeasurement

Desorptionmeasurement

Inerttracer

Main advantage/Maindisadvantage

Appropriate purpose

Batch Chosen(<1 kg L�1)

Easy Difficult (stepby step)

No Easy-to-use/unrealistic forporous media

Preliminarymeasurements

Stirred flowcell

Chosen(<1 kg L�1)

Easy Easy Possiblyno

Flow-through with chosen solid/solution ratio

Reaction kinetics, releasemeasurement

Repackedcolumn

High(Pbulkdensity)

Easy Easy Yes Solid/solution ratio of porousmedia/structure destroyed

Transfer in poorlystructured porous media

Zero-lengthcolumn

High(Pbulkdensity)

Difficult (orfast reaction)

Easy Possiblyno

Negligible dispersion/lowreaction time

Kinetics of release


4.2.2. The flow-through methods

A solution that does not contain the consideredreactive solute is continuously injected at the inletof the reactor. Then, the reactive compound is (i)instantaneously injected at the inlet of the reactor(Dirac injection) or (ii) continuously injected duringa time period (finite step injection) or (iii) continu-ously injected after a given time (infinite step injec-tion). The outlet solution can be flushed or re-injected (Maraqa, 2001a). In addition, one canadjust the flow rate, so that the mean residence timecan be chosen to be higher or close to the meanreaction time, or even lower if the reaction is notinstantaneous. Thus, the kinetic dependence is eas-ier to investigate than in the batch method.

An excellent review has been written about therelations between sorption isotherms and break-through curves (Schweich and Sardin, 1981) andsome of the simple rules were formally establishedby Golden (1969). A recent review provides someother examples of relations between isotherms andbreakthrough curves (Seidel-Morgenstern, 2004).In the common case of a concave isotherm, theasymmetry of the breakthrough curve is strength-ened. However, it is important to remember thatmany other mechanisms can create asymmetry, suchas (i) the presence of immobile water (e.g. VanGenuchten and Wieranga, 1976, 1977; Van Genuch-ten et al., 1977; Valocchi, 1985; Ma and Selim, 1997;Maraqa, 2001b), or (ii) slow adsorption or desorp-tion kinetics (e.g. Rubin, 1983, or see Fig. 5), or(iii) preferential flowpaths (Jury and Roth, 1990;Jury and Fluhler, 1992), or (iv) colloidal transport(Ryan and Elimelech, 1996; Kretzschmar et al.,1999), or (v) simply a low ratio between advectionand dispersion (low Peclet number). So, the experi-

ments must be performed in different conditions(injected concentrations, flow rates, etc.) in orderto test their influence independently and to avoiderroneous conclusions. In the presence of noticeableasymmetry, the main experimental difficulty consistsin measuring precisely the tail (Schweich and Sar-din, 1981; Altfelder et al., 2001; Maraqa, 2001a;Pang et al., 2003). Most of the time, the recoveryof the adsorption or desorption isotherm from abreakthrough curve needs to solve a system contain-ing the transport equation of the inert tracer addedto an equilibrium and/or a kinetic model of theadsorption/desorption. This mathematical issue isoften complex. However, one can provide two sim-ple qualitative rules: (i) if two adimensioned break-through curves of a reactive tracer made with twodifferent injected concentrations and the same flowrate are not superimposed, it means that the iso-therm is not linear; and (ii) if two adimensionedbreakthrough curves of a reactive tracer made withtwo different flow rates and the same injected con-centration are not superimposed, it means that akinetic effect is involved. More details about therecovery of an isotherm from a column or a stir-red-flow breakthrough curve are discussed in thefollowing paragraphs.

4.2.2.1. The stirred flow-through reactor. This systemis also named ‘‘stirred flow cell’’ or ‘‘stirred flowchamber’’ and was developed following the conceptof the continuous-flow stirred tank (Carski andSparks, 1985; Rimstidt and Dove, 1986; Randleand Hartmann, 1987; Miller et al., 1989a; Schnabeland Fitting, 1989; Seyfried et al., 1989; Sparks,1989; Zhang and Sparks, 1993) that is widely usedin engineering and industry. It consists in injecting


a step of the reactive tracer through a stirred cellcontaining a known mass of the solid in contactwith the solution, and then comparing the break-through curves of the reactive solute with that ofan inert tracer.

For a perfect stirred flow-through reactor, thesolution composition is assumed to be homoge-neous within the reactor and equals the outlet one.The breakthrough curve of an inert tracer followinga step injection from C = 0 to C0 (Fig. 12b) is givenby Eq. (26) (Denbigh, 1944; Villermaux, 1993)

C ¼ C0 1� exp�uV 0

ðt � t0Þ� ��

ð26Þ

where t is time, t0 is the instant when the tracer isinjected, / (L s�1) represents the flow rate and V0

(a) Batch

stage 2: eq

stage 1: fast adsorption

C0 = 10-4 mol L-1

(b) Stirred flow-through reactor

C0 = 10-4 mol L-1C0 = 10-4 mol L-1

0 1 10

maximum

median

minimum

C0 = 10-4 mol L-1

0.4

0.3

0.2

0.1

C/C

0

adsorption stage

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

V/V0

C/C

0

Time (

Fig. 12. Adsorption kinetics of Co on a soil measured (a) in batch ansolution (from Limousin, 2006). The batch method indicated a fast abecause of long-term side effects, such as particle disrupting. On the cmean water residence times confirmed the existence of slow reactions.

(L) is the volume of the solution inside the reactor(i.e. the volume of the reactor minus the volumeof the solid). The ratio V0// defines the mean resi-dence time in the reactor for an inert tracer. Forthe desorption stage (switching back the injectedconcentration to zero), the breakthrough curve ofan inert tracer (Fig. 12b) is given by Eq. (27) (Den-bigh, 1944; Villermaux, 1993)

C ¼ C0 exp�uV 0

ðt � t1Þ� �

ð27Þ

where t1 is the instant when the injection of the tra-cer is stopped.

Knowing these two equations, it is often not nec-essary to measure the breakthrough curve of aninert tracer: most of the time, there is no difference

stage 3: adsorption enhanced by particle disrupting ?

uilibrium or slow adsorption

g

100 1000

C0 = 10-4 mol L-1

desorption stage

14 16 18 20 22 24 26

inert tracercobalt, mean water residence time = 1 hcobalt, mean water residence time = 20 h

h)

d (b) in stirred flow-through reactor with the same backgrounddsorption, but it prevented from concluding to a slow reactionontrary, the stirred-flow through method used with two different


between the computed and measured breakthroughcurve when the solid is well mixed. For a reactivesolute, the breakthrough curve deviates from thatpredicted in Eq. (26) (resp. (27)) and the amountof adsorbed (resp. desorbed) solute is given by thesurface area separating the breakthrough curve ofthe reactive solute and of the inert tracer (Fig. 13).Modelling the breakthrough curve of a reactive sol-ute consists in solving a system of several equations:Eq. (26) or (27) coupled with an equilibrium or akinetic model (e.g. Bar-Tal et al., 1990 for someanalytical solutions taking into account a kinetic

Fig. 13. Construction of the adsorption isotherm from the breakthrougflow-through reactor, assuming that the reaction is instantaneous. Thecurve and the solid concentration Qi is given by: Qi ¼ flow rate

solid mass� Ai ¼

isotherm is obtained by the same procedure with the desorption stage

dependence). The difference between the surfacearea of the adsorption stage and the surface areaof the desorption stage provides the amount of reac-tive solute which was pseudo-irreversibly adsorbed(since the quantity of adsorbed solute can be negli-gible compared with the total injected quantity, thistype of mass balance is often more appropriate toknow the amount of pseudo-irreversibly adsorbedcompound than the difference between the totalinjected mass and the total recovered mass). With-out coupling the chemical-transport model, it is pos-sible to construct the isotherm by discretizing the

h curve of an inert and a reactive tracer obtained with the stirredrelation between the hatched surface area Ai of the breakthroughflow rate

solid massðR t¼ti

t¼0Cinert tracer dt �

R t¼ti

t¼0Creactive compound dtÞ. The desorption

of the breakthrough curve.


studied concentration range (from C = 0 to C0) inton concentration intervals (the ith interval being fromC = 0 to Ci). Then, the corresponding adsorbedfraction between C = 0 and Ci is calculated usingthe surface area between the breakthrough curvesof the inert and reactive tracers inside this interval(Fig. 13). Each of these calculations provides onepoint (Ci,Qi) of the isotherm. This method has theadvantage of avoiding any model assumption andrequiring only one breakthrough curve to constructa complete isotherm. However, several issuesremain. First, a precise isotherm requires a highnumber of points. But the greater the concentrationintervals are, the smaller is the precision on eachsurface area between the breakthrough curves ofthe inert and reactive tracers. In addition, thegreater the non-linearity of the isotherm, the smallerthe difference between two intervals must be inorder to have good precision, which leads to thestrong assumption that the reaction must be consid-ered instantaneous not only compared to the meanresidence time of the inert tracer, but also to thetime corresponding to each concentration interval.These limits in the precision of this method explainwhy the coupling-transport models are often pre-ferred. Martin-Garin et al. (2003) showed, however,the feasibility of constructing the isotherm by per-forming several breakthrough experiments with dif-ferent concentrations, each of them providing onlyone point of the isotherm by the total mass balancebetween the reactive and the inert tracer, instead ofseveral points for one breakthrough curve. Thismethod is valid only if the isotherm can be consid-ered linear in the concentration range of each break-through curve, but it provided good consistencywith other batch data (Fig. 11).

The two main disadvantages often pointed out inthe literature are (i) sealing, which prevents con-ducting long experiments and (ii) the magnetic stirbar placed inside the cell that progressively destroysthe solid particles (Deitsch and Smith, 1995; Pigna-tello, 2000). These two problems can be avoided (i)by reversing the sense of the flow inside the cellintermittently without reversing the complete circuit(Heyse et al., 1997; Barthes et al., 2004) to preventthe filters from being sealed, and (ii) by using anexternal system that shakes the cell instead of plac-ing a stir bar inside (Furrer et al., 1993; Zysset andSchindler, 1996).

In conclusion, the stirred flow-through reactor ismuch more appropriate than the batch method tostudy the release stage (e.g. Grolimund et al.,

1995; Martin-Garin et al., 2003; Szenknect et al.,2005), and provides satisfactory results to predictthe breakthrough curves obtained with a repackedcolumn (Szenknect et al., 2005). Moreover, the flowrate (and thus the mean contact time of the solutewith the solid) can be adjusted to measure the influ-ence of chemical kinetics (e.g. Bar-Tal et al., 1990;Eick et al., 1990; Nagy and Lasaga, 1992; Van Cap-pellen and Qiu, 1997a,b; Martin-Garin et al., 2003),as shown in Fig. 13 (Limousin, 2006).

4.2.2.2. The repacked column. This column is packedwith the solid particles, and the solution goesthrough. An example of typical breakthroughcurves of inert and reactive tracers is shown inFig. 5 (Limousin, 2006). Like the stirred flow-through reactor, this method has the advantage ofbeing an open-flow method. So, the chemical kinet-ics and the release stage can be studied more easilythan with the batch method (Sparks et al., 1980;Sparks and Rechcigl, 1982). Moreover, the solid/solution ratio is representative of natural porousmedia. Furthermore, the experiments can be runeither in water saturated or unsaturated conditions(Plassard et al., 2000; Porro et al., 2000). Howeverthis flow-through method has some disadvantages.For example, the duration of the experiment canbe long, especially for strongly adsorbed solutes orin the case of compacted of unsaturated clay mate-rials with low hydraulic conductivity.

The main disadvantage of this technique is thatthe system is not perfectly mixed and the break-through curve of the reactive compound also con-tains the hydrodynamic dispersion of the medium.So, a comparison with an inert tracer is unavoidableto know how chemical reactions are involved.Taking into account hydrodynamic processes addsto the complexity of the problem and, contrary tothe stirred-flow through method, it prevents con-structing the isotherm without choosing a chemicalmodel. Some analytical solutions for the case ofnegligible dispersion (De Vault, 1943; Glueckauf,1945) have shown their limitations because ofthe dispersion or of the presence of immobilewater (Glueckauf and Coates, 1947; Merriam andThomas, 1956). Thus, fitting a coupling chemical-transport model to the breakthrough curve and thenverifying the parameters of the model in differentconditions remains the simplest method to obtaininformation on the adsorption or desorptionisotherm. Moreover, several numerical models havebeen developed to calculate the adsorption or


desorption isotherm from dispersive breakthroughcurves (e.g. Schweich et al., 1983; Kool et al.,1989; Griffioen et al., 1992).

4.2.2.3. The zero-length column. An original methodhas been proposed to nullify the effects of the por-ous network (hydrodynamic dispersion, preferentialpathways, immobile water, etc.). The column is thinenough so that its dispersivity is close to zero andthe breakthrough curve is only sensitive to thechemical properties of the medium (Eic and Ruth-ven, 1988; Ruthven and Eic, 1988).

This method is particularly suited to highlyadsorbed compounds or to desorption studies(especially for already polluted porous media).Compared with classical column methods, thistechnique can be performed without an inert tra-cer. However, the length of the column must bethin enough and the flow rate must be high enoughto overcome the dispersive effects. Thus, thismethod is often not suitable for adsorptionmeasurements.

5. Conclusion

The sorption isotherm is a common approach todescribe a great diversity of retention/release phe-nomena. This is very useful and often unavoidableto understand and predict the mobility of sorbingsubstances in the environment. However, thisapproach is macroscopic and empirical in nature,thus not saying, by itself, anything on the compli-cated mechanisms involved. In particular, it isimportant to verify if thermodynamic equilibriumis reached within the reaction- (or residence-) time,both for the retention and for the release stage ofthe compound. Otherwise kinetic experiments mustbe considered.

Since the isotherm is not an intrinsic property ofthe substance/solid couple, the measurementmethod has a great influence on the results. Thus,it must be chosen carefully and always describedwith the results in detail.

Other methods allow the investigation of theretention microscopically, particularly with spectro-scopic and microscopic tools. They have provided anew efficient way to verify several assumptions usedin isotherm interpretations on the solid structureand retention/release mechanisms, thus leading tomore confidence in structure-based and mecha-nism-based complexation models. These modelsshould replace the traditional ‘‘Kd’’ approach. On

the other hand, the increasing power of computersmakes possible not only improving mechanisticmodels of speciation, but also running ‘‘MolecularDynamic Experiments’’. However, natural mediaare such complicated mixtures of numerous mineraland organic compounds that empirical approachessuch as the ‘‘sorption isotherm’’ will be still usedfor a long time.

Acknowledgements

The authors acknowledge partial funding from‘‘Electricite de France’’ and French Atomic EnergyCommission. We thank all the reviewers who partic-ipated in the improvement of this paper.

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