Ž.Geoderma 99 2001 225–243 www.elsevier.nlrlocatergeoderma Description of sorption data with isotherm equations Christoph Hinz) Soil Science and Plant Nutrition, The UniÕersity of Western Australia, Nedlands,WA 6907, AustraliaInstitute of Soil Science and Forest Nutrition, UniÕersity of Gottingen, Busgenweg 2, ¨¨ D-37077 Gottingen, Germany ¨ Received 3 December 1999; received in revised form 25 May 2000; accepted 26 May 2000 Abstract Analysis of sorption data is important for characterizing retention of chemicals by soil. In this paper I review the most common isotherm equations used in soil science. The mathematical descriptions of these equations are classified into rational, power, and transcendental functions which are related to the isotherm classification. I use most well-known isotherm equation as special cases of a general equation. The meaning of the parameters of this equation is discussed in terms of the Giles classification. Guidelines are presented for choosing the correct type of isotherm equations to describe a set of sorption data. In particular, I show that plotting the distribution coefficient vs. the amount of solute sorbed to the solid phase on a log–log scale is the best way to identify the class and subgroup of isotherms based on the Giles classification. Examples of how to apply the guidelines to determine and modify isotherm equations are presented. q2001 Elsevier Science B.V. All rights reserved. Keywords: sorption; isotherms; equations; soils; solutes 1. Introduction Sorption isotherms are widely used to characterize retention of chemicals in soils. The importance of using isotherm equations is reflected by their incorpora- ) Fax: q61-8-9380-1050.E-mail address: [email protected] ŽC. Hinz . . 0016-7061r01r$ -see front matter q2001 Elsevier Science B.V. All rights reserved. Ž. PII: S0016-7061 00 00071-9 From a mathematical point of view.. As yet. I summarize a number of well-known isotherm equations into a general isotherm equation. and transcendental functions. Isotherm classification and mathematical description 2. because the goodness of fit criteria obtained from parameter estimation often suggests how well an equation describes the data. points of inflection. Classification after Giles et al. 1994.. Simple equations. they further distinguished between subgroups in each class shown in rows 5 to 7 of Fig. 1. First. choose an equation that will describe the data with a given accuracy. Finally. I apply this method to two data sets and illustrate how isotherm equations can be modified to fit data over a wide range of concentrations. isotherm equations can be grouped into rational. Hence. and maxima. One prerequisite is that we know in advance which isotherm equation may describe the data. Langmuir Ž L. A simple inspectional method is introduced to determine the types of isotherms and equations that are suitable to describe the data. we need to employ structural analysis. Consequently. regression methods have to be employed to find an optimal parameter set. 1990. Ž1974. I review isotherm equations which have proven to be useful for describing sorption data..1. Often these equations do not accurately describe the data. . . . An adequate description of sorption data therefore requires two steps. constant partition Ž C. transport models for assessing the mobility of chemicals often require isotherm parameters as input. Lack of an accurate description of sorption data may yield serious errors when applied to transport modeling ŽHinz et al. . using the isotherm classification scheme suggested by Giles et al. . Ž 1974. 2. They distinguished between high affinity Ž H. more complicated expressions have to be used ŽKinniburgh. power. (1974) Giles et al.226 C. no inspectional analysis exists that allows researchers to easily determine the type of isotherm equation that best fits a specific data set. For determining functional relationships between two quantities. and sigmoidal-shaped Ž S. Webster Ž 1997. such as MINTEQA2 ŽAllison et al. 1. isotherm classes Ž Fig. classified sorption isotherms based on their initial slopes and curvatures. Also. These equations are classified. To account for plateaus. Secondly. In this paper. such as the Freundlich or Langmuir isotherms are commonly used to describe sorption data. Hinz r Geoderma 99 (2001) 225–243 tion into chemical speciation programs. . reviewed the use of regression in soil science. 1986. These two steps are of course not completely independent. This phenomenological classification is based on pure observation and does not . it is crucial that sorption data are described adequately with appropriate equations. . Regression methods are usually employed to determine the parameters of isotherm equations. . Ž1974. shown with differently transformed coordinates.C. Isotherm classification according to Giles et al. Hinz r Geoderma 99 (2001) 225–243 227 Fig. 1. 228 Table 1 Compilation of isotherm equations. and their classification according to Giles et al. L4 S2 L3 L3 Equation lim c C. Hinz r Geoderma 99 (2001) 225–243 ™ 0 dc ds lim c ™` s lim s ™`c Class ž ž ž 1q k 2 c sT k1c /ž /ž / 1y k 3 c / / / sTŽ f1 k1q f 2 k 2 . k 2 q sT k1y k 2 r k 3 1q k 1 c X y sT k1 Xc q q 1q k 1 c 1q k 1 c k2 c k2 = y c 1y k 2 c k3 ž ž / . Name Rational functions Langmuir Two-site Langmuir Modified Langmuir Brunauer–Emmett–Teller Farley–Dzombak–Morel sT sT sT kc 1q kc f1 k1c 1q k 1 c k1c 1q k 1 c k1c f2 k2 c 1q k 2 c 1 1q k 2 c 1 sT k q sT sT sT na na na na na 1r k 3 1r k 2 L2 L2. Ž1974. 0 k1 Ž X y sT . their properties at high and low concentrations. kc 1q kc kc / a 1q Ž kc . Hinz r Geoderma 99 (2001) 225–243 L2 Ž kc . 229 . L2. a a a ` sT sT sT sT na ` na na na na L1. kc sT k Ž 1q Ž kc .Power functions Freundlicha General Langmuir Freundlich General Freundlich Redlich–Peterson Toth ´ Aranovich–Donohue Transcendental functions Temkin Fowler–Guggenheim KFca sT sT sT sT ` Ž0. H1 S2 a For a ) 1 Freundlich equation belongs to S1 and has zero slope at c s 0 Žlisted in parenthesis. H2 L2 C. c ` sT k ` sT ` na L1. . ž k1c 1q k 2 c /ž 1 1y k 3 c / k1 1r k 3 L3 k 1 ln c q k 2 sr sT 1y s r s T s k 1 expŽ k 2 s r s T . ` ` sT k a 1r a a ž 1q Ž kc .. H2 L2. H1 ŽS1. . Generally. or at high sorption concentration Žlim s ™ ` c . we need to know which classes of isotherms are most important for soils and. Ž1974. Type IV is equivalent to the L4 and H4 isotherm since it has two .. mathematical criteria must be used. Type I is characterised by a constant sorption maximum and a convex shape. This is especially true for low concentrations. 1985. Finally. s. on the other hand. Subgroups are defined by sorption behavior at high concentrations. use a roman numbering from I to VI for their classification. Giles et al. whereas subgroup 2 is characterized by one plateau. A different classification of sorption phenomena of gasrsolid systems is given by the International Union of Pure and Applied Chemistry ŽSing et al. Subgroup 1 shows no plateau. expressed as a straight line in s vs. subgroup max is defined by the existence of a maximum Žnot shown in Fig. we need to take into account which processes lead to particular isotherm types. Ž 1974. slope at low concentration Žlim c ™ 0 d srd c .. C isotherms are defined by a constant sorption affinity. asymptotic behavior at high solution concentration Ž lim c ™ ` s . Hinz r Geoderma 99 (2001) 225–243 reveal the processes that lead to different isotherm shapes. Ž1985. I provide mathematical criteria which include curvature Ž convex or concave given by d 2 srd c 2 evaluated at a point of interest. It is therefore equivalent to the L2 and H2 isotherms of Giles et al. This indicates that the sorption affinity of H isotherms increases with decreasing concentration. K d plots are useful for distinguishing between different classes and subgroups. Subgroup 3 has an inflection point due to a change from a plateau to a concave shape. and they used qualitative criteria lacking mathematical formalism.. used classification to describe data rather than equations. Type II exhibits an inflection point at which the shape changes from convex to concave similar to the L3 and H3 isotherms. 1. 1 summarizes these criteria and shows the plots of K d s src vs.. the slopes of H isotherms reach high values whereas slopes of L isotherms remain constant.230 C. shows that K d values at c s 0 equal the slope of the isotherm at c s 0 by taking the limit using l’Hopital’s rule: lim K d s lim c c ™0 ™0 c s s lim c ™0 d c ds Ž1. Sing et al. The previous criteria for characterizing isotherm equations are included in Table 1. While both H and L isotherms have a convex shape. S isotherms have a concave shape at low concentrations.. Two plateaus are characteristic of subgroup 4 isotherms.. and the number of plateaus and inflection points. c plots. The distribution coefficient K d is often used to transform sorption data to identify parameters for L-type isotherm equations. On one hand. Ž 1. To apply this classification to isotherm equations. Fig. Eq. A concave isotherm is classified as type III and has its equivalent in the S1 isotherm of the Giles classification. 1. Ž 1974. the slope at zero concentration is defined by: lim K d s lim c c ™0 ™0 d c s ` ds Ž3. They classified isotherms according to Giles et al. An extensive list of isotherms of organic compounds in soil is provided by Weber and Miller Ž 1989. may satisfy the condition in Eq. L-class isotherm Mathematical criteria to define L-class isotherms are based on the sorption behavior at low concentrations. While the classification of Sing et al. The Langmuir. and showed that C. Furthermore. Mathematically. two-site Langmuir. Finally type VI isotherms belong to the L and H classes with multiple plateaus. H-class isotherm Giles et al. all rational functions with parameters greater than zero meet the above criteria. Generally. 1980. However. A concave isotherm with a sorption maximum is referred to as type V. classification is more suitable for the general description of sorption isotherms especially when the retention processes are unknown. is specifically tailored towards gasrsolid adsorption. . In addition to different shapes. Functions involving power relations. Ž2. according to how they behave at low concentrations. discussed the interpretation of this classification and its significance for sorption of chemicals to soils and presented examples for each group. .2. Ž 1985. subgroup 3 and 4 isotherms are seldom observed. Toth. L4 isotherms were found for nickel sorption by soil from a B horizon Ž Harter. it also takes into account different types of hysteresis loops that are caused by capillary condensation in pores.. ´ and Redlich–Peterson isotherm equations can describe L-type data Ž Table 1. Ž1974. L.C. and Scheidegger fulvic acids and to oxides as presented by Cernık et al. . subgroup 3 and 4 isotherms that belong to either the L or S class are often observed for proton binding to humic and ˇ ´ et al. characterized high affinity sorption isotherms Ž H-class. Hinz r Geoderma 99 (2001) 225–243 231 plateaus. Examples for subgroup 2 isotherms of inorganic solutes can be found in Kinniburgh Ž1986.1. Ž1994. which is the same as the S2 isotherm. Ž1995. 2. such as the Toth ´ and the Redlich–Peterson isotherm equations. Also. ds Ž2. 1983. .1. . the Giles et al. Sposito Ž1984. This classification is based on physical adsorption of gases measured in porous or on dense solid materials. sorption of phosphorus on aluminum oxide was described by L3 isotherms Ž van Riemsdijk and Lyklema.. The slope of L-isotherms should be constant when the concentration approaches zero: lim K d s lim c c ™0 ™ 0 d c s const. and H isotherms are most common.. 2. Second.1. At trace concentrations many solute–soil systems behave this way. a heavy metal may form a stable complex with a chelate. All of these involve power functions which can be derived from continuous site–affinity distributions Ž Kinniburgh. Compared to the L and H isotherms the S-class occurs less frequently. Furthermore. ´ 1995. I will show that the Giles classification and most of the isotherm equations listed in Table 1 are fully compatible with mathematical properties of rational functions. the sorption of a solute may be inhibited by a competing reaction within the solution. Methylene blue sorption on cristobalite was also described by an S isotherm Ž Burgisser. .. Ž 3.. 1990. . However. 1986. S-class isotherm Isotherms of the S-class have two causes. and the adsorption of ¨ surfactants follows sigmoidal-shaped isotherms Ž Rea and Parks. Cu sorption on a clay loam was attributed to the ligand effect ŽSposito. the Fowler–Guggenheim equation is a S2 isotherm. 2. Most of the commonly used isotherm equations are indeed rational functions derived from the theory of adsorption processes.1.. In many cases the total amount of a solute is measured and not the activity of the different species. especially when the Langmuir equation is an appropriate model for the sorption processes. Proton binding to oxide surfaces may be described by S-class isotherms Ž Scheidegger et al. 2. H-class isotherm equations can be used for solely descriptive purposes. Hinz r Geoderma 99 (2001) 225–243 An affinity that approaches infinity when the concentration reaches zero does not make sense from a thermodynamic point of view ŽToth. Measuring metal concentration with atomic absorption spectroscopy will indiscriminately detect free and chelated species. solute–solute attractive forces at the surface may cause cooperative adsorption which leads to the S-shape ŽGiles et al.2. and the general Freundlich equations. C-class isotherm C-class isotherms exhibit constant affinity for a wide range of concentrations.. Transcendental functions are also used and can describe . 2. For example. 1974.232 C. 1984. The isotherm thus describes the total amount of metal in solution vs. when a sorption data set can be described by an equation that satisfies Eq. 1994. Mathematical description First. a second class of equations involving power laws can efficiently describe a wide range of isotherm types. and is therefore a multicomponent effect. the general Langmuir–Freundlich. the amount sorbed to the solid phase. such as a complexation reaction with a ligand.3. S-class isotherms can be described by the Freundlich equation with a power greater than one and by the modified Langmuir equation. implying that K d is constant. Many organic substances follow a C1 isotherm. Also..4. 1994. . First. Examples of this type of isotherm equation are the Freundlich. sorption data approach an asymptotic sorption maximum. the number of sites n is also the highest power of polynomials in the nominator and denominator of the rational function. s T is a parameter indicating asymptotic sorption at high concentrations Ž mol kgy1 soil. although. Ž4. Ž 4. . rational functions can describe asymptotic behavior at high concentrations. can be viewed as a rational function if a common denominator can be found and the powers a i j . . Ž4. is used for pure descriptive purposes.C. and f i is the fraction of site i with Ý n is 1 f i s 1. . It is not by chance that most isotherm equations listed in Table 1 are rational functions. Ž4. and g i j are dimensionless empirical parameters. for example.. Eq. Sums of Langmuir equations can be written as rational functions. 2. Rational functions are known to be superior over polynomials for interpolation or extrapolation Ž Press et al. I will first consider the equation where all powers are equal to one. and the exponents a i j . 1992. It should be emphasized that the parameters are empirical coefficients when Eq. In this paper sites are defined as different types of reactive surfaces that may be described with their own sorption relationship. bi j . Rational functions Rational functions are quotients of polynomials. which is essentially a rational function. all coefficients of rational functions are greater then zero. Also. and g i j are equal to one. a polynomial is not able to describe the maximum without introducing fluctuations around this asymptotic value. which relate the mathematical structure of Eq. they are seldom used. In order to relate well-known isotherms to Eq.2. where s denotes the amount of solute sorbed to the solid phase Žmol kgy1 soil. to the Giles classification. Only for very well-defined systems can parameters be interpreted as meaningful system properties. bi j . we first need to discuss the relationships between the parameters and characteristics of the isotherm and the possible meaning of the parameters. This implies that discrete site–affinity distributions of Langmuir affinity parameters can always be written as rational functions. Ž4. The parameters pi j and q i j are empirical coefficients Ž moly1 l. The total number of sites is denoted by n and m i gives the number of interaction terms..1. Unlike simple polynomials. If. . Ž4. summarizes how parameters of Eq. and c is the solute concentration in aqueous solution Žmol ly1 . I have written all isotherm equations in Table 1 as sums and products of simple Langmuir type expressions. Hinz r Geoderma 99 (2001) 225–243 233 some of the sorption phenomena in soil. need to be adjusted to obtain isotherm equations of Table 1. An equation describing all classes requires a flexible expression of the following form: n mi s s sT Ý fiŁ is 1 js 1 ž qi j c a i j 1 q pi j c b i j gi j / Ž4.. To better understand Eq. In particular. general Freundlich. Ž4. modified Langmuir.a -1 1 0 . showed that rational function models exhibit Aclose-to-linearB behavior meaning that the asymptotic behavior is reached with . These properties are only asymptotically approached for least-squares estimators of nonlinear models Ž Ratkowsky. However. Redlich–Peterson. showed that rational functions can describe a wide variety of ascending and descending curves. b p 11 s 0 required for FRD equation. Apart from their utility for interpolating mathematical functions. 1987. Freundlich.1 1r a 1 1 )0 1 1 1 1 1 1 1 1 1 1 1 1 1 )0 )0 )0 )0 )0 TSL. Least-squares estimators of model parameters are unbiased and normally distributed when the model is linear and the error is unbiased and identically normally distributed. two-site Langmuir. FDM. rational functions also have statistical properties that makes them suitable for efficient parameter estimation. BET. ML. FRD.234 C. to obtain well known isotherm equations Rational functions with bi j sg i j s 1 Name a n i sT Langmuir TSL ML BET FDM 1 2 1 1 4 1 1 2 1 1 1 2 3 4 Power functions with qi j s pi j ) 0 Name a n i sT FRD GLF GF Toth ´ RP AD c a b mi 1 1 1 2 2 1 2 2 1 j 1 1 1 1 2 1 2 1 1 2 1 2 1 qi j )0 )0 )0 )0 1 )0 1 )0 /0 )0 )0 )0 )0 pi j q11 q11 q11 q11 )0 q11 -0 q11 )0 -0 )0 -0 0 ai j 1 1 1 1 0 1 0 1 1 0 1 0 0 )0 )0 )0 )0 )0 mi 1 1 1 1 1 2 j 1 1 1 1 1 1 2 ai j )0 0 . general Langmuir–Freundlich. Hinz r Geoderma 99 (2001) 225–243 Table 2 Parameter adjustment of Eq. Ratkowsky Ž1987. Farley–Dzombak–Morel. c p 12 . AD. . RP. GF. Brunauer–Emmett–Teller. This means that in general a large number of data points is required to apply nonlinear least-squares techniques. Ratkowsky Ž1987. GLF. Aranovich–Donohue.a -1 1 1 0 bi j 1 a 1 a )0 1 1 gi j 1 1 0 -g .0 and q12 s 1 required for AD equation. Ž4. and Farley– Dzombak–Morel Ž FDM.0. with pi j ) 0 and qi j ) 0. Power functions Isotherm equations that involve power functions are often based on the Langmuir equation. and g i j are derived from averaging local-affinity coefficients that are continuously distributed at the microscopic scale ŽSips. and Toth ´ isotherms ŽTable 1. f i can be viewed as the fraction of total surface area which each site i constitutes. the parameter s T equals the plateau for isotherms of subgroup 2 of H and L isotherms. The Freundlich isotherm with a ) 1 can be derived from the general Langmuir–Freundlich isotherm for low concentrations ŽSposito. For multisite isotherms Ž n ) 1. takes place Ž Table 1. such as the Gauss–Newton method. This interpretation of s T is only valid for very well defined sorbents and sorbates. Isotherms which are concave at high or low solution concentrations require that m i ) 0. 1950. 1938. Farley et al. the exponents a i j .. general Freundlich isotherm. describes a class S isotherm. Also. or precipitation Ž FDM. . The asymptotic behavior at high s values can be interpreted as the concentration of 1rqi j at which either surface condensation ŽBET. Sposito.. For pi j ) 0 and qi j . When isotherm equations are derived from surface reactions s T relates to the surface area. . For sums of simple isotherms. such as multisite Langmuir equations. Eq. I have written Eq. 1984. we have Ý n is 1 f i s 1. isotherms are examples of L3 class isotherms Ž Brunauer et al. isotherms of subgroup 4 with more than one plateau require n ) 1. 1990. In addition to isotherms that are based on continuous site affinity distributions. In particular. In particular. exponents may be viewed as heterogeneity factors which normally vary between 0 and 1.. the distribution of the local affinity coefficients becomes wider.. Nederlof et al.. Table 1. classification of isotherms. With increasing qi j . The parameters qi j s pi j are positive real numbers. where the plateau concentration of site i is equal to f i s T .C. With a decreasing value of the exponent. in terms of sums and products because this is the easiest way to make it compatible with the Giles et al. We can therefore describe isotherms of the S class and subgroup 3 of the L and H classes. Hinz r Geoderma 99 (2001) 225–243 235 a small number of data points. bi j . will converge rapidly to an optimal set of parameters. . 1984. The Brunauer–Emmett–Teller ŽBET. Ž4. . 2.. such as a modified Langmuir isotherm Ž Schmidt and Sticher. The exponent g i may be greater than 1 if m i ) 0 and qi j s pi j . It can also be viewed as a measure of the affinity of a solute to sorb to a surface. the slope becomes steep and the solution concentration rapidly approaches the plateau of L2 and H2 isotherms. the equation describes isotherms belonging to subgroup 3 where lim s ™ ` c s 1rqi j . . This ensures that least-squares techniques.2. The affinity coefficient qi j is related to an equilibrium constant for adsorption to well defined surfaces. 1985.2. 1986. Special cases of isotherm equations with m i s 0 are the general Langmuir–Freundlich. which means that two hyperbolic functions are multiplied. In this context. In fact. 3.2. 1981. which is similar to the presentation of Giles et al. since s is dependent on c. they are of little importance for describing sorption data in soil. The Fowler–Guggenheim equation describes adsorption where interactions between adsorbed solutes occur so as to form S2 isotherms ŽAdamson. 1990. Transcendental functions Transcendental function can also be used to describe sorption data Žsee also Table 1. Ž1974.. . to the data. . it is a measure of sorption affinity. c on a linear scale. Another transcendental function is the Dubinin–Radushkevich equation that was used and modified by Kinniburgh Ž1986. I suggest that K d be plotted vs. 1. if bi ) a i . This yields a larger scatter of data points in K d vs. Once the class and subgroup are determined. c. Outliers which may cause problems during parameter estimation can easily be detected by plotting data this way. one can use an isotherm equation from Table 1. This plot reveals the type of isotherm. Identifying sorption isotherm equations 3.. the Temkin isotherm can be used to describe binding-site heterogeneity ŽStumm and Morgan. plots of K d vs. Plots of sorption data on a linear scale will show a concentration range within two orders of magnitude. . s or c plots. In fact. Because sorption data are often determined for concentrations ranging over several orders of magnitude. This can be done by plotting the data in different ways. s or c. it is useful to first assign the class and subgroup of the classification of Giles et al. Nonetheless. High affinity sorption isotherms Ž H. .. are employed. Because K d is defined as src. require exponents so that the slope will approach infinity when c approaches zero. In particular. sometimes referred to as Freundlich plots. Data within this range can therefore be plotted as s vs. . one can describe isotherms of subgroup 3 and 4 with exponents greater than one. one can still obtain a maximum. Although isotherms of subgroup max seem to be rare for natural soils.3. 2. A qualitatiÕe approach to determining sorption equations To determine which equation best describes any given sorption data.1. Ž1974. The general Freundlich and general Langmuir–Freundlich isotherms belong to the H class Ž Table 1. Instead. s or c amplify variations of sŽ c . as shown in Fig. He showed that a modified version of this equation has similar properties to the Toth ´ isotherm. log–log plots of s vs. log–log plots are not suitable to identify class and subgroup of isotherms by inspection. However. Hinz r Geoderma 99 (2001) 225–243 the Redlich–Peterson isotherm has a fractional power based on interactions of adsorption sites.236 C. Furthermore. Accordingly. As pointed out by Veith and Sposito Ž 1977. this scheme helps to determine whether an initial amount of solute is sorbed to the soil as shown by a maximum in log K d vs. log s plot Žitem Ž5. A detailed description of how to determine parameters of the two-site Langmuir equation is given by Sposito Ž1982. have to be adjusted to account for certain phenomena in the log K d vs. showed that a two-site Langmuir can fit any convex sorption data that tends to a maximum asymptotically. Sposito Ž 1982. Also. This scheme should be viewed as a qualitative tool in finding an appropriate equation.. . I have developed a scheme which classifies a given data set Ž Table 3. demonstrated how one can analyze sorption data with . 1. this plot is especially useful to determine whether or not a two-site Langmuir equation is suitable to describe the data. it is also evident how the parameters in Eq. In fact. .. other isotherm equations as listed in Table 1 should be used. s reveals this behaviour. parameter estimation should be carried out with untransformed data. At low sorption concentrations. 1984. To systematically determine isotherm class and subgroups. Kinniburgh Ž1986. parameters of the two-site Langmuir equation can be determined from the slopes of the data points at high values of K d and from the slope at low values of K d Ž Sposito. is given by log K d s 1 log K F q Ž a y 1. Plotting K d vs. which is not commonly used. Subgroup 3 isotherms show a minimum at high s values. this plot is more sensitive at low concentrations than other linearizations. While this linearization provides insight in the choice of the correct isotherm equation. s on a log–log scale will yield a straight line for Freundlich isotherms. From Fig.. log s plots. K d vs. Furthermore. Langmuir-type isotherms exhibit constant K d which appears as a horizontal line in logarithmic Scatchard plots. H-isotherms have negative slopes whereas S-isotherms have positive ones at low concentrations. Classification of isotherms into subgroups depends on isotherm behavior at high concentrations. .C. Ž4. s plots are referred to as Scatchard plots and they transform the Langmuir isotherm into a straight line ŽSposito. 1982. This linearization reveals when sorption data deviate from the Freundlich equation at low concentrations and hence aid in choosing the correct isotherm equation to describe the data. : K d s ks T q ks Ž5. a sorption maximum for subgroup 2 isotherms appears as asymptotic behavior at low K d values. This linearization of the Freundlich isotherm. a a log s Ž6. If both equations are inadequate for data description. A simple inspection of the trend of K d vs. Hinz r Geoderma 99 (2001) 225–243 237 On a linear scale. It is especially useful for determining whether or not a Freundlich or Langmuir isotherm is suitable to describe sorption data. asymptote 2.0 1. Fig. slope at low s 1.14 to 413 mg ly3. convex line Žlinear at low c. curvature at low c Properties 1. c plotted on a linear scale.2 orders of magnitude Plot s vs.1. but low concentrations are not shown at a resolution that makes it possible to recognize the class of isotherm. negative slope 2. c Curve Ž1. To determine whether the Langmuir equation is appropriate. sŽ0. We need therefore to plot the data on a log–log scale as indicated by criteria A and A) in Table 3. straight line 2. straight line Ž5. Hinz r Geoderma 99 (2001) 225–243 Table 3 Qualitative approach to identify isotherm class and subgroup and sorption model Data range ŽA. convex line Žconvex at low c. Plots of K d vs. s on a linear and log–log scale exhibit a maximum at low s values.238 C. Examples To show how Table 3 helps to determine an appropriate isotherm equation. 3. Sorption of 1. . equivalent to three orders of magnitude ŽTable 4. c . c ) 2 orders of magnitude log K d vs. 2c. I present two examples.3-dinitrobenzene on montmorillonite was investigated at concentrations ranging from 0. s in Fig. K d is plotted vs. this isotherm belongs to the S class or the soil . an equation with more than two parameters needs to be employed to describe the data. . s ŽA) . S class L class H class S2 type isotherm any class with constant term Subgroup 2 Subgroup 3 K d vs. negative 1. 2a shows the data s vs. 1. s 0 2. 4. behavior at high s an initial sorption by subtracting an adjustable parameter from the isotherm equation. The convex shape is clear. Following Table 3 A) item Ž 5. positive slope Ž4. sŽ0. straight line with negative slope 1. zero 3. Since the data points do not follow a straight line. concave line 3. Freundlich isotherm Ž a ) 1. positive 2. minimum Isotherm C class S class L class H class Langmuir isotherm Freundlich isotherm Ž a . maximum at low s Ž6. log s Ž2. Fig. The curvature indicates that the Freundlich isotherm is not appropriate to describe the data.2. 2b shows the same data on a log–log scale. curve over the whole data range Ž3. 62 0.. Eq.9 k2 Žmmoly a la . Ž7. s on a linear Žc. and unmodified and modified Aranovich–Donohue equations ŽAD and mAD. 0. 2 shows the best fit. and a logarithmic scale Žd. contained an initial amount of 1.51 a1 a2 1.55 0. and without a constant Žsolid line. Ž8. General Langmuir–Freundlich equation was fitted to the data with Ždashed line.36 0. – – 3.33 7. Eq.3-Dinitrobenzene Benzene Benzene a mGLF GLF mAD AD 0. Apersonal communicationB.3 69. It systematically overestimates data points at low concentrations suggesting that the model is not Fig.39 Parameters are from unmodified and modified general Langmuir–Freundlich equations ŽGLF and mGLF. Adsorption of 1. The dashed line in Fig.3-dinitrobenzene on montmorillonite saturated with KCl. The general Langmuir–Freundlich equation was used in a first attempt to describe the data. 1997.3-dinitrobenzene and the regular and modified Aranovich–Donohue equations for benzene sorption on dry Kettleman loam Example Equationa sT Žmmol kgy1 . y44.013 38.81 – – – 0. and a logarithmic Žb. .3-Dinitrobenzene 1.C. 2. Hinz r Geoderma 99 (2001) 225–243 239 Table 4 Results of parameter estimation of the best fit of the original and modified general Langmuir– Freundlich equation for 1...0076 0. 2600 2316 – – s0 Žmmol kgy 1 .. scale and K d vs. c on a linear Ža. Ždata are from Fesch.3-dinitrobenzene. At high concentrations the data seem to reach a sorption maximum indicating a subgroup 2 isotherm. Data are plotted s vs.1 – – – k1 Žmmoly a la . 3 shows the best fit which systematically underestimates the data at low concentrations. s on a linear Žc. Hinz r Geoderma 99 (2001) 225–243 adequate at these concentrations. Data points lie within a range of two orders of magnitude. and the original Žsolid line. s is plotted on linear and log–log scales. c r denotes normalized concentration with respect to the condensation concentration. It shows that data points of K d increase with decreasing s and therefore that K d is not constant at low concentrations. A modified Ždashed line. The solid line in Fig. Ž 1993. scale and K d vs. The Aranovich–Donohue Ž AD. . 3a and b show s vs. Adsorption of benzene on Kettleman soil ŽShonnard et al.240 C. 3. where s0 denotes a constant Žmmol kgy1 . c plots on linear and log–log scales. They measured sorption of benzene on dry Kettleman soil. Fig. 1993. The solid line shows the best fit which describes the isotherm over the whole range of concentrations equally well. Data are plotted s vs.. We know that a maximum in a K d vs. 1995. c on a linear Ža. and a logarithmic scale Žd. The general Langmuir–Freundlich equation is then written as s s sT k 1 c a1 1 q k1c a1 q s0 Ž7. and a logarithmic Žb. After an initial convex shape the isotherm becomes concave and seems to approach an asymptotic value on the c axis implying that this isotherm falls into either the L or H class and subgroup 3. This is even more pronounced in Fig.. equation describes an L3 isotherm ŽAranovich and Donohue. Aranovich–Donohue equation was fitted to the data. . . s plot can be described by adding a constant to the isotherm equation.. . 3c and d where K d vs. Fig. The second example is taken from Shonnard et al. 4. I used an unweighted nonlinear least-squares technique to obtain an optimal set of parameters. In this paper I have demonstrated that the appearance of sorption data can be related to the mathematical form of isotherm equations by using the Giles et al.C. where a 1 is an additional parameter which makes the first part of this equation identical to the general Langmuir–Freundlich equation. Hinz r Geoderma 99 (2001) 225–243 241 To account for this behavior I modified the Aranovich–Donohue equation by introducing a power function as follows: ss ž k 1 c a1 1 q k 2 c a1 /ž 1 1 y k3c a2 / Ž8. 1986. Regularization techniques were used to determine affinity spectra. classification. 1978. Holford. One purpose of using regularization is to incorporate a priori knowledge into parameter optimization routines. They addressed the suitability of specific isotherm equations for describing sorption phenomena and statistical issues of curve fitting. power. Persoff and Thomas. 1996. 1988. in the context of discrete and continuous addressed by Cernık site–affinity distributions. Schulthess and Dey. Harter and Baker. 1977. Inspectional analysis presented . 1978. An inspectional analysis of sorption data using this classification and graphical representations of the data has been shown to be a useful tool in identifying isotherm class and subgroup. Kinniburgh. Conclusions and outlook Parameter estimation of isotherm equations has been addressed by several authors ŽVeith and Sposito. Eq. Two examples have been presented to show how inspectional analysis can be used to modify isotherm equations. Harter. The description of the data at low concentrations has improved considerably without systematic deviations. 3. The asymptotic behavior at low and high concentrations of these equations is most useful for determining the class and subgroup of isotherms. 1977. reveals the structure of the mathematical expressions in isotherm equations. Ž1995. Mathematical expressions can be grouped into rational. Statistical issues of how over-parameterized equations can be used to describe sorption data have been ˇ ´ et al. The best fit is shown as dashed lines in Fig. The problem of describing sorption data when both Freundlich and Langmuir equations fail to describe data was not resolved.. Ž 4. and transcendental functions. The difference between the original AD equation and the modified AD equation is that at low concentrations the modified equation belongs to the H instead of the L class. Knowledge of the relations between the isotherm equation and data appearance reveals which group of isotherm equations of Table 1 should be used. The first two functions are most useful. 1984. . Hinz r Geoderma 99 (2001) 225–243 in this chapter provides qualitative a priori knowledge on the mathematical description of data. 106. D. 1077–1080.. Am.0. H. 1995.A. Koopal. 41.242 C. Sci. E. 5th edn. M.D. 175. a geochemical assessment model for environmental systems. 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