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Abstract

Chemosphere 58 (2005) 115–123

Short Communication

www.elsevier.com/locate/chemosphere

Modiﬁcation of Langmuir isotherm in solution systems—deﬁnition and utilization of concentration dependent factor

a, b * Seungman Sohn , Dongsu Kim

a Department of Chemical Engineering, Virginia Tech., Blacksburg, VA 24061, USA b Department of Environmental Science and Engineering, Ewha Womans University, Daehyundong 11-1, Seodaemungu, Seoul 120-750, Korea Received 25 November 2003; received in revised form 15 July 2004; accepted 27 August 2004

The Langmuir isotherm, originally derived for the adsorption of gas molecules on solid surfaces, was modiﬁed to ﬁt the adsorption isotherm of solutes onto solid surfaces in solution systems. The aim of this modiﬁcation is based on the fact that direct application of the Langmuir isotherm to solution systems often leads to poor data ﬁtting. In the present communication, it is shown that the level of data ﬁtting to the Langmuir isotherm of literature data can be improved by a simple modiﬁcation introducing a concentration dependent factor,X. The key concept of the modiﬁcation lies in that the concentration of solute aﬀects both adsorption and desorption stages. As a ﬁrst approximation, we adopted a single-term polynomial for both processes of adsorption and desorption. Based on reanalysis of literature data of adsorption in solution, we conﬁrmed that indeed the modiﬁed Langmuir isotherm more accurately describes the experimental observations. Furthermore, we proposed that the concentration dependent factor could be associated with the surface heterogeneity index that was introduced in a few other modiﬁed Langmuir isotherms. Some advantages and limitations of proposed modiﬁed Langmuir isotherm are also discussed. 2004 Elsevier Ltd. All rights reserved.

Keywords:Langmuir isotherm; Adsorption; Desorption; Solute concentration; Surface heterogeneity

1. Introduction

Adsorption of gas or solute molecules on solid sur-faces has been a critical point of interest both in scien-tiﬁc aspects and in industrial applications in the ﬁeld

* Corresponding author. Present address: E-Ink Corpora-tion, 733 Concord Avenue, Cambridge, MA 02138, USA. Tel.: +1 617 499 6180; fax: +1 617 499 6200. E-mail address:ssohn@eink.com(S. Sohn).

of environmental science. For instance, development of eﬃcient catalysts for removal of metal ions in solution system must be based on a thorough understanding of adsorption isotherms at diﬀerent temperatures and con-centration (Boudart and Djega-Mariadassou, 1984). Re-cent environmental concerns also drive the advent of more eﬀective molecular sieves and selective membranes by which economical and commercially viable gas sepa-ration is feasible (Riazi and Khan, 1999). Adsorption phenomenon in solution systems also plays an impor-tant role in many areas of practical environmental

0045-6535/$ - see front matter2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chemosphere.2004.08.091

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S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

technology. In water and wastewater treatment, for example, the adsorption technique has found wide appli-cability because of several advantages such as high eﬃ-ciency, simple operation, and easy recovery/reuse of adsorbent. Today, new types of adsorbents are con-stantly being developed and as a result, the realm of adsorption technique is not restricted only to environ-mental science (Eckenfelder, 1989; Davies et al., 1997; Oezer et al., 1997; Sujana et al., 1998; Saeed et al., 1999), but also expands to soil (Campbell, 1995; Huang and Cheng, 1997; Altin et al., 1998; Singh and Pandeya, 1998), corrosion (Fontana and Staehlel, 1970; Subrama-nian and Lakshminarayanan, 2002), and microbiologi-cal sciences (Oezer et al., 1999; Docoslis et al., 2001). The characteristics of the adsorption behavior is gen-erally understood in terms of both equilibrium (i.e., ther-modynamic viewpoint) and adsorption kinetics. For kinetics, temporal variations of the amount of adsorp-tion are measured and thus obtained experimental data are used to develop a proper kinetic model. If similar sets of data as a function of temperature are available, activation energy for adsorption can be determined from an Arrhenius-type equation. For equilibrium aspects of adsorption, thermodynamic parameters such as changes in Gibbs free energy, entropy, and enthalpy can be eval-uated from adsorption isotherms. In this type of study, the construction of an adsorption isotherm plays a key role in understanding the adsorption mechanism. Nor-mally, the studies of thermodynamics of adsorption allow us to judge the feasibility of a particular process, while kinetics studies ﬁnd the most eﬃcient way of oper-ating the process. To understand the adsorption isotherm, the Lang-muir equation is perhaps the most widely used model due to its simplicity and strong theoretical reasoning be-hind. Three essential premises of the Langmuir isotherm are monolayer coverage, adsorption site equivalence and independence. In general, these premises are well ob-served for gas adsorption on homogeneous surfaces (Bawn, 1932). It is recognized, however, that heteroge-neity of solid surfaces plays a dominating role in adsorp-tion from the gas phase on many adsorbents of industrial utility (e.g., activated carbons, silica gels, aluminum oxides, etc) (Ross and Oliver, 1964). Several attempts were made to describe single-gas adsorption on heterogeneous solids with noticeable theoretical understanding (Jaroniec and Brauer, 1986). It is of more current interest to develop the theoretical foundation for predicting mixed-gas adsorption equilibria by means of parameters that characterize single-gas adsorption systems. In contrast to gas phase adsorption, if we look into the nature of solute adsorption in solution systems, the sec-ond and the third assumptions may not hold because sol-utes will have a tendency to adsorb onto more active sites where pre-adsorbed molecules, such as water molecules,

could be easily displaced. This type of heterogeneous adsorption often leads to an island adsorption, which could be, at least in part, the origin of unsatisfactory data ﬁtting by the Langmuir isotherm. In this work, we pro-pose a modiﬁed Langmuir isotherm by which the level of data ﬁtting in solution systems can be signiﬁcantly im-proved. During modiﬁcation we assumed that, in solu-tion systems, solute concentration aﬀects not only the adsorption stage but also the desorption step. To simplify the model we adopted a single-term polynomial in the description of the solute concentration dependence in both adsorption and desorption stages. The modiﬁed Langmuir isotherm has two advantages compared with other isotherms. First, in contrast to Freundlich isotherm that is widely known to better ﬁt the solution adsorption data with two adjustable parameters, the modiﬁed Lang-muir isotherm has only one variable. Second, as will be detailed later, the magnitude of this parameter may pos-sibly be related to the level of surface heterogeneity. We anticipate that the newly proposed model may ﬁnd its application and further our understanding in the studies of adsorption in solution systems.

2. Modiﬁcation of Langmuir isotherm

2.1. Langmuir isotherm

In any given adsorbate-adsorbent system, dynamic equilibrium between adsorption and desorption of adsorbate,X, (e.g., gas or solute) onto an adsorbent, M, can be expressed as follows.

XþMsurface¡XMðSurfaceÞ

ð1Þ

From this simple equilibrium, one can deﬁne a few important parameters: surface coverage (h) deﬁned as the ratio of the number of adsorption sites occupied to the number of available sites, and adsorption (ka) and desorption (kd) rate constants. Based on these parame-ters, the rate of change of surface coverage (dh/dt) dur-ing adsorption step will be given as:

½dh=dt ¼;hÞ adskafaðC

ð2Þ

wherefais an adsorption function depending on the con-centration of adsorbate (C) and the surface coverage. Following the same logic we can also formulate a similar equation for the desorption stage.

½dh=dt ¼kdfdðC;hÞ des

ð3Þ

in whichfdis a desorption function. At (dynamic) equi-librium, [dh/dt]adsand [dh/dt]desmust be equal, which leads to the following general form of isotherm.

½fðC;hÞ=fðC;hÞ ¼k d a eq a=kd

ð4Þ

S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

Now let us consider the Langmuir isotherm that is a special form of Eq.(4). Langmuir postulated that the exact shapes offa(C,h) andfd(C,h) are equal to, respec-tively,CÆ(1h) and (h) which further simpliﬁes Eq.(4) as follows.

h¼KC=ð1þKCÞ

ð5Þ

in whichCis the concentration of gas molecules (i.e., pressure) andKis deﬁned aska/kd. As mentioned before, the Langmuir isotherm is based on the three key assumptions: monolayer coverage, sites equivalence and sites independence. These premises may be oversim-pliﬁed but in the case of gas adsorption on a well-deﬁned solid surface, they are reasonable assumptions (Bawn, 1932).

2.2. Modiﬁcation of Langmuir isotherm

In solution systems, the adsorption of solutes on solid surface may not hold the above three premises. For instance, some authors (Kim and Chung, 2001) note 2 poor Langmuir ﬁtting,Rvalues between 0.37 and 0.85, upon the modeling of orthophosphate adsorption in titanium dioxide. Two reasons underlie why Langmuirs original isotherm for gaseous systems is potentially prob-lematic to describe the adsorption in solution systems. First, when a species is adsorbed from solution, there should be accompanying desorption of another species for charge-balance considerations (Harter and Baker, 1977). Second, the existence of surface heterogeneity (i.e., a distribution of surface energy of adsorbent) may lead to an island-type adsorption, which clearly deviates from the important Langmuir premise of adsorption site equivalence. A good example would be the process of wastewater treatment in which activated high surface area substrates such as activated carbon are used to extract heavy metal ions. In this regard, to be able to apply the Langmuir isotherm in solution sys-tems, some modiﬁcations are necessary. The modiﬁcation of the Langmuir isotherm must start from Eq.(4), and the key issue here is to deter-mine the exact forms offa(C,h) andfd(C,h) in solution systems. In the present study, as a ﬁrst approximation, we made two hypotheses. First, in the adsorption proc-ess, solute follows a similar type of adsorption behavior as in gas adsorption but with more general behavior de-m scribed by a polynomial term,CÆ(1h). Second, the solute concentration will also aﬀect the desorption stage. At this moment we cannot determine the exact mathematical shape offd(C,h); yet to take the simplest polynomial function we also assumed it is equal to n CÆh. This leads us to a modiﬁed Langmuir isotherm as follows.

X X h¼KC=ð1þKCÞ

ð6Þ

117

whereCis solute concentration, andX( =mn) is an exponent indicating the level of concentration depend-ence. Details regarding the nature of the exponent,X, will be given in the Discussion section. Eq.(6)can be re-shaped to a more practical form in which the surface coverage (h) is expressed as the ratio of adsorbed amount of solute (q) at speciﬁc equilibrium condition to maximum adsorption (qmax)). Therefore, by substitut-ingq/qmaxforh, and by taking the reciprocals on both sides of Eq.(6), the following equation is obtained. X 1=q¼1=ðKqÞx½1=C þ1=qð7Þ max max X From a linear regression of 1/qversus [1/C] , the val-ues ofKandqmaxcan be determined, respectively, from the slope andY-axis intercept. It needs to be noted that the exponentXis a ﬁtting parameter by which we can determine the best linear regression for a given system. The validity of this modiﬁed Langmuir isotherm will be tested in the following section along with the discus-sion of its limitations.

2.3. Literature data revisited

As an example of gas adsorption on homogeneous solid, we chose the system of CO adsorption on freshly cleaved mica at 193 K. The actual data points were reproduced from the literature (Bawn, 1932) and reana-lyzed according to Eq.(7). In 1932 Bawn clearly speci-ﬁed that the main purpose was to conﬁrm the importance of the Langmuir isotherm and to throw fur-ther light on the nature of the monomolecular adsorbed ﬁlm. The data in this reference has been reproduced in many current physical chemistry books as a typical example of Langmuir isotherm (Noggle, 1985).Fig. 1 shows the variation of 1/V(Vis volume of gas) as a X function of [1/Cwhere the concentration dependent] , factor,X, varies between 0.4 and 1.8. In the insert, the variation ofXand the corresponding least square ﬁt 2 (R) were listed. Consistently the closed symbols show Xvalues less than 1.0 and the open symbols represent Xgreater than 1.0. Data ofXis denoted by the= 1.0 cross symbols. As clearly seen, asXvalue increases the 2 data ﬁts better, andRreaches its maximum point at XAbove= 1.0. Xthe degree of ﬁtting becomes= 1.0, 2 poor. InFig. 2,RversusX, in the system of CO adsorp-tion on mica surface at two diﬀerent temperatures of 90 K and 193 K was shown. It is apparent that, regard-less of adsorption temperature, the best linear ﬁt, repre-2 sented by the highestR, is obtained whenXis equal to 1. Above or below the dotted line showingX= 1, 2 Rdecreases from its maximum value. An example of adsorption in solution systems is con-trasted inFig. 3showing the adsorption of lead ions ontoSchizomeris leibleinii, a kind of green alga, at 30C and pH = 4.5 (Oezer et al., 1999). The same nota-tions as shown inFig. 1are used. In this case, the

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S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

Fig. 1. A plot of linear Langmuir isotherm (reciprocal volume versus inverse gas concentration) of CO adsorption on Mica surface at 193 K (actual data points have been reproduced from the literature (Bawn, 1932) and reanalyzed using Eq.(7)). The exponent,X, derived upon the modiﬁcation of Langmuir, is a curve ﬁtting parameter quantiﬁes the inﬂuence of solute concentration in adsorption isotherm.

2 Fig. 2.Rfrom the least square ﬁt versusXin the system of CO adsorption shown inFig. 1. The dotted line showsX= 1 where the modiﬁed Langmuir equation shown in Eq.(7)turns to the form of original Langmuir isotherm.

2 maximum value ofRoccurs atX= 0.6, which is signif-2 icantly lower thanX= 1.0 (see the insert ofXversusR).

2 Fig. 4carries the information ofRversusXfor three diﬀerent metal ions-iron, lead and cadmium-adsorbed

S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

119

Fig. 3. A plot of linear Langmuir isotherm of lead ions ontoSchizomeris leibleinii, a kind of green alga, at 30C and pH 4.5 (actual data points have been reproduced from the literature (Oezer et al., 1999) and reanalyzed using Eq.(7)).

1.000

0.975

0.950

0.925

0.900 -0.3

0.0

0.3

0.6

Iron Lead Cadmium

0.9

1.2

1.5

1.8

2 Fig. 4.Rfrom the least square ﬁt versusXshown inFig. 3. The dotted line showsX= 1 where the modiﬁed Langmuir equation shown in Eq.(7)turns to the form of original Langmuir Isotherm.

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S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

2 Fig. 5. (a)Rfrom the least square ﬁt versusXin the study of adsorption isotherms of caesium ion in various Welsh soils (actual data 2 points have been reproduced from the literature (Campbell, 1995) and reanalyzed using Eq.(7)). (b)Rfrom the least square ﬁt versus Xin the study of the eﬀect of selected azoles, such as benzimidazole (BIMD), mercapto benzimidazole (MBIMD), imidazole (IMD), and benzotriazole (BTA) on the growth of oxide ﬁlm on copper surface in 0.1 M NaOH solution (actual data points have been reproduced from the literature (Subramanian and Lakshminarayanan, 2002).

on the same adsorbent of alga at optimum conditions in 2 each system. The maximumRin iron and cadmium sys-2 tems occurs aroundX= 1.0. The overall shape ofRver-susXfollows a similar trend as shown inFig. 2.

2 More examples ofRversusXfrom the adsorption in solution systems are presented inFig. 5(a) and (b), the data of which are based on two references, (Campbell, 1995) for (a) and (Subramanian and

S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

Lakshminarayanan, 2002) for (b). Campbell studied adsorption isotherms of caesium ion in various Welsh soils using the Langmuir model. Some of their work was reproduced inFig. 5(a). The numbers in the figures legend designate a speciﬁc location where the topsoil was sampled. Clearly various soils having diﬀerent origin showed a wide range of concentration dependence be-tween 0.7 and0.15. Of more interest, none of the sam-ples showed anXvalue close to 1 indicating the soil surfaces might be highly heterogeneous. A potential relationship between the magnitude ofXvalue and the surface heterogeneity will be detailed in the Discussion section. Subramanian and Lakshminarayanan (2002)studied the eﬀect of some of selected azoles, such as benzimidaz-ole (BIMD), mercapto benzimidazole (MBIMD), imi-dazole (IMD), and benzotriazole (BTA) on the growth of oxide ﬁlm on copper surface in 0.1 M NaOH solution (Subramanian and Lakshminarayanan, 2002). Their point of study was to understand the mechanism of passivation and adsorption behavior of various azoles on copper surface. InFig. 5(b), some of their results 2 were reanalyzed in terms ofRandX. Most of the azoles studied (BIMD, MBIMD, and BTA) exhibit adsorption isotherms best represented by the unmodiﬁed Langmuir 2 equation (i.e.,X= 1 at highestR). The only exception is 2 IMD that showsXat= 0.4 RThe authors= 0.973. mentioned that they tried to ﬁt the data using various isotherms, and they concluded that the Langmuir equa-tion ﬁts the best. Although the details of inhibition mechanism by these azoles are beyond the scope of this paper, one important correlation between the adsorp-tion of azoles and the eﬃciency as an inhibitor may be found. The authors stated that, based on the results of cyclic voltametry, IMD showed theleastinhibition (inhibitor performance, BIMDMBIMD > BTA > IMD). As a reason for this, they proposed that the ab-sence of phenyl ring unlike in other azoles, IMD might inherently possess less ability of inhibition. In this work, we further claim that in addition to diﬀerent chemical nature of IMD, more heterogeneous, thus island-type, adsorption on copper surface judging from theXvalue far deviating fromX= 1 may be at play for the unsatis-factory inhibition eﬃciency.

3. Discussion

In the present work, the key concept of a Langmuir isotherm modiﬁcation is to assign more freedom both in adsorption and desorption stages in terms of adsorb-ate concentration so that experimental data in solution systems can ﬁt better. During the modiﬁcation, two important cautions must be noted. First, the number of independent adjustable parameters should be minimized. Second, each parameter must possess a physical meaning

121

more than simply being a curve ﬁtting parameter. From this point of view, the current model has some excellence in that it has only one ﬁtting parameter that quantiﬁes the concentration dependence of adsorption. At this moment it would be necessary to compare the current model with preexisting adsorption isotherms such as the Freundlich isotherm (Freundlich, 1926). The Freundlich isotherm was originally derived based on the assumption that most surfaces are heterogeneous so that the surface coverage,h, will be a function of both solute concentration and heat of adsorption (Q) (i.e., break down of the premise ‘‘site equivalence’’ in Lang-muir isotherm). In this case, the measured adsorption isotherm,H(C,T), now, may be expressed as follows. Z HðC;TÞ ¼fðbÞ hðC;b;TÞdbð8Þ

wheref(b) is the distribution function for the heat of adsorption in overall adsorption sites, in which 0 b=bexp(Q/RT), andh(C,b,T) is the adsorption iso-therm. Most of the solutions of Eq.(8)are thoroughly discussed in the monograph byJaroniec and Madey (1988). As a special case, assuming that the distribution of heat of adsorption follows a Gaussian function (i.e., f(Q) =aexp(Q/nRT), whereais a pre-exponential con-stant), we obtain the Freundlich adsorption isotherm. 1=n H¼aCð9aÞ or 1 lnðHÞ ¼lnðaÞ þnlnðCÞ ð9bÞ 0 where a equals toaRTnb, and 1/n, the slope in a double logarithmic plot, is the intensity of adsorption (nP1). From Eq.(9), it is clear that the Freundlich isotherm has two independent ﬁtting parameters (aandn). It also should be noted that the Freundlich isotherm, unlike the Langmuir, lacks a solid thermodynamic reasoning, bur rather serves as an semi-empirical equation. We can also ﬁnd other modiﬁed isotherms that are primarily based on the Langmuir isotherm. A common feature among various modiﬁed isotherms is to ﬁt better the adsorption data obtained on heterogeneous surfaces using the Langmuir isotherm. The ﬁrst eﬀort to extend the Langmuir model to adsorption on an energetically heterogeneous solid was exercised by Sips in 1948. Based on the assumption of quasi-Gaussian energy distribu-tion, Sips derived an isotherm containing the surface heterogeneity factor,b, as follows. b b hðpÞ ¼ ðApÞ=½1þ ðApÞ ð10Þ wherepis an equilibrium pressure of adsorbed gas mol-ecules, and is a Langmuir constant containing the char-acteristic adsorption energyUo(i.e.,A¼AoexpðUo=RTÞ, whereAois a pre-exponential factor). Following Sips pioneering work, many similar adsorption isotherms containing surface heterogeneity factor were developed.

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S. Sohn, D. Kim / Chemosphere 58 (2005) 115–123

Examples would be Redlich-Petersen (Yong et al., 1992) and Dubinin-Raudshkevich (Iwata et al., 1995) iso-therms. Once again, these isotherms have a similar form as shown in Eq.(10), but with minor diﬀerences in con-stant and the use of surface heterogeneity factor,b. A close examination of Eq.(5), (9a) and 10reveals that the form of Sips isotherm (Eq.(10)) resembles the com-bination of Langmuir (Eq.(5)) and Freundlich (Eq. (9a)) isotherms. The term 1/n in the Freundlich model indicates the intensity of adsorption, which, in turn, must be ascribed to the distribution of heat of adsorp-tion (Q) or surface heterogeneity. In this regard, the exponent 1/n per seincubates a comparable physical meaning to surface heterogeneity factor,b, used in Sips and other Langmuir modiﬁcations. In the present work, the modiﬁed isotherm in Eq.(6) resembles Sipsisotherm. However, it should be clearly noted that the derivation and mathematical treatment of these two equations are entirely diﬀerent. We solely considered the concentration dependence during adsorp-tion and desorption stages. The Sips model and other modiﬁcations are based on a Gaussian distribution of adsorption energy, which correlates to the surface hetero-geneity of adsorbent. The nature of the exponentXin modiﬁed Langmuir isotherm (Eq.(6)) also needs to be fully evaluated. WhenX= 1, the modiﬁed isotherm re-turns to the original form of the Langmuir isotherm. WhenXis smaller than 1, the particular adsorbate-adsorbent system exhibits less concentration dependence; while,Xis greater than 1, the system experiences higher concentration dependence. In other words, to obtain the same level of surface coverage (or adsorption amount), the system ofX< 1 requires less amount of solute, while theX> 1 system mandates more. Although the above statement is qualitative, the concentration dependent fac-tor,X, may have a plausible correlation withb. If the sur-face of given adsorbent is heterogeneous, one may anticipate an island type of adsorption where the free en-ergy of adsorption is more negative. In this case, accord-ing toSips (1948),bbecomes smaller than 1. In parallel, a heterogeneous surface may exhibit less concentration dependence since high-energy surface area will be prefer-entially covered even at low solute concentration. There-fore, heterogeneous surface (i.e.,b< 1) is coincident with the system that has less dependence on solute concentra-tion (i.e.,X< 1). In the following, we suggest another sup-port showing thatXandbexhibit similar behavior. By deﬁnition,bcannot be greater than 1 (i.e.,b= 1 means a completely homogeneous surface, perhaps as exempli-ﬁed inFig. 1). However,Xcan have any value, as long as it ﬁts the adsorption data. In this work we extensively searched literature data covering a wide range of subjects, and to our knowledge, we found that theXvalue that yields the best linear ﬁt is always equal to or less than 1. This excellent correlation needs to be underscored since the two approaches shown in Eqs. (6) and (10) are inde-

pendent, and thus this consistency may support the valid-ity of the newly proposed model. The strong advantages in Eq.(6)are its simplicity and versatility. For any adsorp-tion systems characterized in terms of adsorbate concen-tration and adsorption amount, we can apply Eq. (6) and deﬁne the degree of concentration dependency, which, in turn, may be correlated to the surface heterogeneity,b. Before concluding this section it is necessary to con-sider some limitations inherent in Eq.(6). First, in the derivation we simpliﬁed the assumption that solute con-centration may aﬀect both adsorption and desorption stages by choosingfa(C,h) andfd(C,h), respectively, as m n CÆ(1h) andCÆh. This is only an approximation, and the exact solution may not be easily obtained since the solute concentration (C) and surface coverage (h) would be coupled with each other (see Eq.(8)for a gen-eral form of surface coverage). Second, if the concentra-tion dependence factor,X, indeed has an intimate correlation withb, the window ofXshould be between 0 and 1 (0 for extremely heterogeneous, and 1 for an ideal homogeneous surface). However, as we observe in Fig. 5,Xbecomes negative for some cases. For Xthe term (< 0, mn) must be negative (i.e.,m<n), which means the solute concentration eﬀect upon desorption stage is stronger than that in adsorption stage. In reality, this has a weak physical meaning since, on a highly heterogeneous surface, the driving force for adsorption will be considerably greater than that in desorption; therefore, once adsorbed, solutes are antici-pated to exhibit less concentration dependence during the desorption stage. We suggest that the appearance of negativeXoriginates from the curve ﬁtting proce-dure, and again, this put some cautions of usingXas an indicator to compare among diﬀerent systems.

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