# Improper Estimation of Thermodynamic Parameters in Adsorption Studies with Distribution CoefficientKD (qe/Ce) or Freundlich Constant (KF): Considerations from the Derivation of Dimensionless Thermodynamic Equilibrium Constant and Suggestions

Authors: Hai Nguyen Tran
Journal: Adsorption Science & Technology
 (2022)

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## Abstract

Adsorption processes often include three important components: kinetics, isotherm, and thermodynamics. In the study of solid–liquid adsorption, “standard” thermodynamic equilibrium constantKEqo; dimensionless) plays an essential role in accurately calculating three thermodynamic parameters: the standard Gibbs energy change (∆G°; kJ/mol), the standard change in enthalpy (∆H°; kJ/mol), and the standard change in entropy [∆S°; J/(mol × K)] of an adsorption process. Misconception of the derivation of the KEqo constant that can cause calculative errors in values (magnitude and sign) of the thermodynamic parameters has been intensively reflected through certain kinds of papers (i.e., letters to editor, discussions, short communications, and correspondence like comment/rebuttal). The distribution coefficient (KD) and Freundlich constant (KF) have been intensively applied for calculating the thermodynamic parameters. However, a critical question is whether KD or KF is equal to KEqo. This paper gives (1) thorough discussion on the derivation of thermodynamic equilibrium constant of solid–liquid adsorption process, (2) reasonable explanation on the inconsistency of (direct and indirect) application of KD or KF for calculating the thermodynamic parameters based on the derivation of KEqo, and (3) helpful suggestions for improving the quality of papers published in this field.


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## Body

## 1. Introduction

In the studies of adsorption, adsorption thermodynamic plays an important role in estimating adsorption mechanism (physisorption or chemisorption). Adsorption processes in solid–liquid phases actually contain at least three important components: adsorbent (solid), adsorbate (solute), and water (solvent). However, the derivation of the standard thermodynamic equilibrium constantKEqo is explained in many different ways in the literature [1–6]. When KEqo is applied for calculating the thermodynamic parameters (i.e., ∆G°, ∆H°, and ∆S°) of an adsorption process, the standard state conditions of this process must be defined [5, 7]. The misconception of such derivation can lead to misapplying the relevant constants of adsorption equilibrium. As a result, the thermodynamic parameters are calculated without physical meanings. Therefore, it is necessary to give further discussion and considerable concerns on its derivation (KEqo) in the field of adsorption (solid and liquid phases).Recently, there are a remarkably increasing number of publications regarding the application of the distribution coefficient (KD) [8–13] or the Freundlich constant (KF) [14–17] as KEqo for calculating the thermodynamic parameters of a liquid–solid adsorption process. This is because the distribution coefficient obtained at different solution temperatures is often measured very fast and simply in laboratories. Meanwhile, the Freundlich model is more suitable for describing adsorption processes that do not reach a strict plateau region in adsorption isotherms. In essence, adsorption isotherms describe the relationship: qe = f(Ce) [18]. Simultaneously, many scholars give some critical comments on this problem [7, 19–21]. The question is whether the thermodynamic parameters that are calculated based on the distribution coefficient or the Freundlich constant bring physical meanings. Notably, several relevant updated information is also provided herein.In any adsorption study, the amount of substance (adsorbate) adsorbed by solid material (adsorbent) at adsorption equilibrium (qe; mol/kg) is commonly calculated from Equation (1). The units of qe (mol/kg) and Ce (mol/L) are used in this study based on previous suggestions [22–27]. Notably, the thermodynamic parameters need to be reported under the standard states [7, 28]. In this work, the superscript symbol (°) represents the standard states, and the subscript abbreviation “Eq” means equilibrium. For convenience, the maximum adsorption capacity of adsorbent toward adsorbate (mol/kg) is abbreviated as MAC. The adsorption equilibrium constant of relevant model is abbreviated as AEC.
(1)qe=Co−CemVwhere Co and Ce (mol/L) are the concentrations of adsorbate at beginning and after adsorption equilibrium, respectively; m (kg) is the dried mass of adsorbent used; and V (L) is the volume of adsorbate solution.


## 2. Derivation of Standard Thermodynamic Equilibrium Constant

Adsorption process can be considered as a heterogeneous chemical equilibrium [2, 29–31]. Although this definition is always correct for all cases [3], it is acknowledged by the community in the field of adsorption. Therefore, chemical equilibrium between adsorption (→) and desorption (←) processes of adsorbate by adsorbent can be generally expressed as Equation (2) [21]. Detail discussion on this derivation has been reported by Lima and colleagues [21]. Early, some authors have reported other equations principally similar to Equation (2) [1, 32, 33].
(2)AdsorbentSolid+AdsorbateLiquid⇌Adsorbent–AdsorbateSolidIn thermodynamics, the derivation of the thermodynamic equilibrium constant should be started from chemical potentials [2]. However, in this study, equilibrium constant is initially introduced to facilitate tracking by readers. Equilibrium constant is usually defined in the terms of activity (a; unitless) rather than actual concentration in molar [2, 30, 34, 35].The activities of the adsorption sites in adsorbent occupied by adsorbate (aAdsorbent–Adsorbate), vacant adsorption sites in adsorbent (aAdsorbent), and adsorbate in solution (aAdsorbate) are defined in Equations (3), (4), and (5), respectively [5].
(3)aAdsorbent–Adsorbate=Adsorbent–AdsorbateEq×γAdsorbent–Adsorbate(4)aAdsorbent=AdsorbentEq×γAdsorbent(5)aAdsorbate=AdsorbateEq×γAdsorbateIn the book, Crittenden et al. [36] defined “{i} = activity or effective concentration of ionic species, mol/L (M)”. This is because of its definition based on an equation ({i} = γi [i]) similar to Equation (5) by considering “[i] = concentration of ionic species in solution, mol/L (M)”. Ghosal et al. [33] reported the unit of activity was both mol/L and mg/L. However, in chemical thermodynamics, activity must be a dimensionless quantity by definitions and is dependent on the selection of standard states. Therefore, to obtain a strict correction, the concentrations (in molar) of the adsorbent–adsorbate [Adsorbent–Adsorbate], adsorbent [Adsorbent], adsorbate [Adsorbate] need to be reported corresponding to reference states [1, 27, 30, 37, 38]. Because both activity coefficient (γ) and activity (a) are dimensionless quantities (actually unitless) [30, 39], Equations (3), (4), and (5) are expressed as Equations (6), (7), and (8), respectively. The relevant information can be found in reference papers [27, 33], textbook [30], and IUPAC report [40].
(6)aAdsorbent–Adsorbate=Adsorbent–AdsorbateEqAdsorbent–Adsorbate°×γAdsorbent–Adsorbate(7)aAdsorbent=AdsorbentEqAdsorbent°×γAdsorbent(8)aAdsorbate=AdsorbateEqAdsorbate°×γAdsorbatewhere γAdsorbent–Adsorbate, γAdsorbent, and γAdsorbate (dimensionless) are the corresponding activity coefficient of adsorbent–adsorbate, adsorbent, and adsorbate, respectively. The activity coefficient in the solid phase includes γAdsorbent–Adsorbate and γAdsorbent and that in the liquid phase is γAdsorbate.Notably, the concentration of the solid phase of adsorbent–adsorbate [Adsorbent–Adsorbate] needs to be expressed in the term of surface coverage fraction (θ) that is the ratio of the number of occupied (or filled) adsorption sites to the number of total (or available) adsorption sites in adsorbent (Equation (9)). Similarly, the concentration of the solid phase of adsorbent [Adsorbent] is defined as the fraction of available adsorption sites in adsorbent (1–θ; Equation (10)) [1, 3–5, 37, 41].
(9)Adsorbent–AdsorbateEq=NumberoffilledadsorptionsitesNumberoftotaladsorptionsites=θ(10)AdsorbentEq=NumberofvacantadsorptionsitesNumberoftotaladsorptionsites=1−θAfter substituting Equations (9) and (10) into Equation (6) and (7), Equations (11) and (12) are obtained, respectively.
(11)aAdsorbent–Adsorbate=θAdsorbent–Adsorbateo×γAdsorbent–Adsorbate(12)aAdsorbent=1−θAdsorbento×γAdsorbentAccording to IUPAC [42], the activity (a; unitless) is defined as Equation (13). Therefore, the chemical potentials of the adsorbent (μAdsorbent) and adsorbate (μAdsorbate) are expressed as Equations (14) and (15), respectively. When adsorption process reaches equilibrium, the chemical potential of adsorbent–adsorbate μAdsorbent–Adsorbate is given as Equation (16) [2, 34].
(13)a=eμ–μ°/RT=expμ–μ°/RT(14)μAdsorbent=μAdsorbento+RTlnaAdsorbent(15)μAdsorbate=μAdsorbateo+RTlnaAdsorbate(16)μAdsorbent–Adsorbate=μAdsorbent–Adsorbateo+RTlnaAdsorbent–Adsorbatewhere R is the universal gas constant [8.31446 J/(mol × K)] and T is the absolute temperature of adsorption process in the kelvin unit with its symbol of K (kelvin = 273.15 + Celsius) [42]; μ is chemical potential (J/mol or kJ/mol) that is equal to ∆G; μ° is standard chemical potential (equal to ∆G°) depending on the selection of the standard state [28]; and μAdsorbento, μAdsorbateo, and μAdsorbent–Adsorbateo are the standard chemical potentials of the adsorbent, adsorbate, and adsorbent–adsorbate in their standard states, respectively.In essence, Gibbs energy change is initially introduced by American mathematician Josiah Willard Gibbs in the 1870’s; 1839–1903). Although the appellation of the Gibbs “free” energy change has been widely used in the literature, it is not suitable or even not real [43]. The term recommended by IUPAC [42] is “the standard Gibbs energy change” for ∆G°. Therefore, the appellation of “free” is not used in this study. Similar to the Gibbs energy change of reaction (Equation (17)), the Gibbs energy change of adsorption process (∆Gads) is commonly expressed as Equation (18) [2]. By replacing Gibbs energy changes (∆GAdsorbent–Adsorbate, ∆GAdsorbent, and ∆GAdsorbate) by chemical potentials (μAdsorbent–Adsorbate, μAdsorbent, and μAdsorbate, respectively), Equation (18) becomes Equation (19).
(17)ΔGreactionorΔrG=ΔGproducts–ΔGreactants(18)ΔGads=ΔGAdsorbent–Adsorbate–ΔGAdsorbate+ΔGAdsorbent(19)ΔGads=μAdsorbent–Adsorbate−μAdsorbent+μAdsorbateBy substituting Equations (14)–(16) into Equation (19), Equation (20) is obtained.
(20)ΔGads=μAdsorbent–Adsorbateo−μAdsorbento+μAdsorbateo+RTlnaAdsorbent–AdsorbateaAdsorbent×aAdsorbateSimilar to∆Gads, the standard Gibbs energy change of adsorption process (ΔGadso) is defined in Equation (21), so Equation (20) can be expressed as Equation (22) [2, 44]. In this equation, the so-called thermodynamic reaction quotient that is often used under non-equilibrium condition is defined as Equation (23).
(21)ΔGadso=μAdsorbent–Adsorbateo−μAdsorbento+μAdsorbateo(22)ΔGads=ΔGadso+RTlnQ(23)Q=aAdsorbent–AdsorbateaAdsorbent×aAdsorbateWhen adsorption reaches equilibrium, the numerical values ofQ becomes KEqo [2, 44]. In this case, the standard dimensionless thermodynamic equilibrium constant (KEqo; Equation (24)) is similar to the reaction quotient.
(24)KEqo=aAdsorbent–AdsorbateaAdsorbent×aAdsorbateEquation (24) was firstly reported by Graham [32] and then by many scholars [1, 5, 21, 22, 33]. When Equations (8), (11), and (12) are substituted into Equation (24), Equation (25) is obtained as follows:
(25)KEqo=θ/Adsorbent–AdsorbateoγAdsorbent–Adsorbate1−θ/AdsorbentoγAdsorbent×AdsorbateEq/AdsorbateoγAdsorbateThe reference states for pure solvents and pure solids with a mole fraction are unity [30, 34]. According to IUPAC [40, 42, 45], the recommended value for the standard molality related to the standard thermodynamic quantities is 1 mol/kg. Therefore, [Adsorbent–Adsorbate]° and [Adsorbent]° are often assumed to be 1 mol/kg in most studies of adsorption [2, 5]. Meanwhile, the solid-phase activity coefficient (γAdsorbent–Adsorbate and γAdsorbent) can be determined by the Wilson equation [46–51]. For example, in the adsorption study of metal ions (Zn2+, Cu2+, Cd2+, and Na+) using the cationic exchanger Amberlite IR-120, Valverde et al. [47] applied the Wilson equation and the Pitzer limiting law equation to determine the activity coefficient in the solid and liquid phases, respectively. However, for many adsorption cases, the solid-phase activity coefficient is not easy to be correctly estimated by the Wilson equation [51], especially for the case of adsorption isotherm (macroscopic equilibrium). Therefore, many scholars assumed that the activity coefficient of the occupied adsorption sites (γAdsorbent–Adsorbate) is similar to that of the vacant adsorption sites (γAdsorbent) when adsorption process reaches equilibrium [1, 22, 32]. Furthermore, Lin and Juang [35] assumed that the ratio (γAdsorbent–Adsorbate/γAdsorbent) was nearly maintained a constant under same experiment conditions when they applied mass action law (microscopic equilibrium) for determining the thermodynamic equilibrium constant of ion exchange process. As a result of such discussion, Equation (21) can reduce to a simple form (Equation (26)) when adsorption process reaches equilibrium.
(26)KEqo=θ1−θAdsorbateEq/AdsorbateoγAdsorbateIn essence, when adsorption process reaches equilibrium (∆Gads = 0 and Q = KEqo; Equation (22)), the standard Gibbs energy change of adsorption process is directly calculated from Equation (27) [2, 44, 52]. KEqo can be expressed as Equation (28).
(27)ΔGadsocommonlyexpressedasΔG°=−RTlnKEqo(28)KEqo=e–ΔG°/RT=exp–ΔG°RTThe fundamental thermodynamic relation of three thermodynamic parameters (∆G°, ∆H°, and ∆S°) is commonly expressed as follows:
(29)ΔG°=ΔH°−TΔS°Assuming that the changes inΔS° and ΔH° with temperatures are negligible, after substituting Equation (27) into Equation (29), the nonlinear (Equation (30)) and linear (Equation (31)) forms of the well-known van’t Hoff equation (not Van’t Hoff or van’t Hoof equation) [1, 52–55] are achieved. The van’t Hoff equation was initially proposed by the Dutch chemist who is Jacobus Henricus van’t Hoff (1852–1911).
(30)KEqo=exp–ΔH°R×1T+ΔS°R(31)lnKEqo=−ΔH°R×1T+ΔS°RIn adsorption studies, the surface coverage fraction (θ) is often defined as Equation (32) [1, 3, 5, 32, 33, 37, 41]. When [Adsorbate]Eq and [Adsorbate]° are expressed as Ce (mol/L) and C° (mol/L), respectively, Equation (26) becomes Equation (33).
(32)θ=qeQmax(33)KEqo=qe/Qmax1–qe/Qmax×Ce/Co×γAdsorbateor another expression as follows:
(34)qe=QmaxCeKEqoγAdsorbate/Co1+KEqoγAdsorbate/CoCeBased on Equation (34), the constant KL (L/mol) is established as Equation (35) or (36). The well-known Langmuir model is achieved as Equation (37). Equation (36) was recently reported by Lima and co-workers [2]. However, a similar form of this equation has been early published by some other scholars [1, 5, 22].
(35)KL=KEqoγAdsorbateCo(36)KEqo=KLγAdsorbateCo(37)qe=QmaxKLCe1+KLCewhere Qmax (mol/kg) that is the MAC is obtained when Ce→ CS; KL (L/mol) is the Langmuir adsorption equilibrium constant; and Ce (mol/L) and qe (mol/kg) are previously defined in Equation (1). The parameter CS (mg/L or mol/L) is defined as the solute solubility (or the liquid phase concentration at saturation). For example, Seifikar and Azizian [56] reported that the saturation concentration of methyl violet dye in liquid phase (CS) was 33,000 mg/L.Equation (36) describes the relationship between the constant KL (L/mol) of the Langmuir model and the standard thermodynamic equilibrium constant KEqo. The constant KL (L/mol) is more appropriate for calculating the thermodynamic parameters (∆G°, ∆H°, and ∆S°) of the solid–liquid adsorption process with physical meanings. The units of adsorption isotherm that is the plot of qe versus Ce should be presented as mol/kg versus mol/L, respectively [24, 27].Clearly, the magnitudeKEqo (Equations (36)) is not only strongly dependent on the Langmuir adsorption equilibrium constant KL but also C° and γAdsorbate. According to IUPAC [40, 42, 45], “the recommended value for standard thermodynamic quantities is Cө = 1 mol dm–3” (or C° = 1 mol/L). In the adsorption studies of solid–liquid phases, the standard concentration (C°) is 1 mol/L as reported by many scholars [2, 5, 7, 8, 22, 27, 57–61].Some authors suggested the effect of the activity coefficient of adsorbate (γAdsorbate) can be negligible under dilute ionic solution or non-ionic solute at low concentrations [5, 33]. Similarly, Hemond and Fechner [62] concluded that γAdsorbate is very close to 1 within an ionic strength lower than 0.001 M; therefore, its effect can be ignored in freshwater environments. For example, Moreno-Marenco et al. [63] selected γAdsorbate as unitary for the adsorption process of n-butylparaben onto activated carbon within initial adsorbate solutions from 0.10 to 1.03 mmol/L (without the presence of NaCl).In contrast, the activity coefficient of charged adsorbates is commonly calculated according to the (extended) Debye–Hückel equation [8, 35, 37, 64]. Some other methods have been applied to estimate the activity coefficient in liquid phase such as the Pitzer method [35, 47, 51, 64] and Davies equation [35]. Each method often involves some limitations and is effective for a certain range of ionic strengths, so researchers should select what the best method for their investigations is.For example, when ion exchange is a primary adsorption mechanism, Lin and Juang [35] compared the thermodynamic parameters of the adsorption process of Cu2+ ions by two chelating ion exchange resins (Chelex 100 and IRC 748). The authors applied the extended Debye–Huckel limiting law, Davies equation, and Pitzer method to determine the activity coefficient of the cations in the aqueous phase. The result (Table 1) demonstrated that the signs of thermodynamic parameters (∆G°, ∆H°, and ∆S°) calculated based on the activity coefficients (determined by three methods) are the same, and their magnitudes are nearly identical.Table 1
Thermodynamic parameters for the process of Cu2+ adsorption by two resins (Chelex 100 and IRC 748) [35].
T(K)Chelex 100 resinIRC 748 resinKEqo∆G°∆H°∆S°KEqo∆G°∆H°∆S°1. Based on the extended Debye–Hückel equation2880.0025814.2739.6588.110.0028314.0540.8192.832980.0043913.45(y = –4769x+ 10.6)0.0049413.16(y = –4908x+ 11.2)3080.0079312.39(r2 =0.9974)0.0083312.26(r2 =0.9996)3180.0120011.690.0142011.252. Based on the Davies equation2880.0035413.5139.5090.230.0038713.3040.6794.992980.0060212.67(y = –4751x+ 10.9)0.0067612.38(y = –4892x+ 11.4)3080.0108011.60(r2 =0.9978)0.0114011.46(r2 =0.9997)3180.0164010.870.0193010.443. Based on the Pitzer method2880.0038913.2939.9492.520.0042413.0841.4098.242980.0066112.44(y = –4804x+ 11.1)0.0074312.15(y = –4980x+ 11.8)3080.0120011.33(r2 =0.9976)0.0126011.20(r2 =0.9993)3180.0183010.580.0218010.11Note: The unit of ∆G° (kJ/mol), ∆H°(kJ/mol), and ∆S° [J/(mol‧K)]; the information in parenthesis indicating the van’t Hoff equation; and the activity coefficient of the cations in the aqueous phase in KEqo calculated by different methods (using the extended Debye–Hückel equation, Davies equation, and Pitzer method).Notably, all ∆G° values in Table 1 are positive. A question is whether the following conclusion (the adsorption was a non-spontaneous process) can be obtained. In essence, the sign of ∆G is often used as a criterion to evaluate the spontaneity of an adsorption process: (–∆G) spontaneous and (+∆G) non-spontaneous [24, 31, 43, 65–67]. A similar information is found in the textbook [55]: “the sign of ΔrG and not that of ΔrG° determines the direction of reaction spontaneity”. However, in the literature, the spontaneity of adsorption processes is frequently reached based on the sign of ∆G° (Equation (27)) [12, 14, 16, 33, 37, 59, 63, 68–71].Xiaofu et al. [67] suggested that “it may not be appropriate to use ∆G°as a parameter for discussing the spontaneity of a reaction”. They noted “∆G° for a reaction can be either positive, negative or zero” as follows: ∆G° >0 (when KEqo <1), ∆G° <0 (KEqo >1), and ∆G° = 0 (KEqo = 1). Similarly, based on the equilibrium position of a given reaction (i.e., AReactant ⇌ BProduct), Shahwan [31] strongly concluded that “the criterion of spontaneity of sorption must be based on ΔG, not on ΔG°”. Figure 1 shows that ∆G° can be negative, positive, or zero depending on the magnitude of KEqo [31, 38]. Under a given temperature, when KEqo is higher than unity, ∆G° (in Equation (27)) will be negative, and vice versa. Similarly, Lima et al. [66] suggested that negative ∆G° values (because of KEqo>1) imply that Equation (2) tends to proceed in the forward direction (adsorption). In this case, “adsorption is thermodynamically favourable to takes place” [66].Figure 1
Gibbs energy (G) as function of the extent of reaction (ζ) for describing ΔG° (negative, positive, or zero). Adapted from the literature [31] with some modification.Some researchers [3, 5] debated that “ΔG° sign gives no information on the spontaneity of the process in non-standard conditions”. However, they suggested that the spontaneity of adsorption (based on ∆G°) is strongly dependent on the selection of the standard state [3, 5]. For instance, Salvestrini et al. [72] investigated the adsorption process of diclofenac onto activated carbon at different temperatures (288, 298, 308, 298, and 318 K) and reported the positive values of ∆G° (5.8, 7.6, 7.7, and 10.1 kJ/mol, respectively). This is because they selected 1 mg/g for solid phase and 1 mg/L for solute (as the standard states) and applied them for calculating KEqoderived from the Langmuir constant (Table 2). If the selected standard states are 1 mol/kg for solid phase and 1 mol/L for solute, the ∆G° values will be negative (Table 2). The concluded suggested the sign of ΔG° is also strongly dependent on the selected standard state and then the magnitude of KEqo (i.e.,KEqo>1 leading to ∆G° <0; Equation (27). The selection of standard states does not affect the sign and magnitude of ∆H° (Table 2). However, there is not any convincing reason to explain why the authors [72] selected the standard states (1 mg/g for solid phase and 1 mg/L for solute) instead of 1 mol/kg and 1 mol/L. In the literature, other researchers [4] selected the different values for the standard state of adsorbate (i.e., 1 mg/L, 1 g/L, 1 mmol/L, and 1 mol/L; Table 3) when they applied Equation (36) for describing the relationship between KL and KEqo. Another opinion is regarded to the solubility of adsorbate. Saeed et al. [73] selected the standard state of dye C° as the solubility value of methylene blue dye at a given temperature, for example, 43,210 mg/L, 60,010 mg/L, and 83,430 mg/L at 298 K, 308 K, and 318 K, respectively. They explained that the value C° (1 mol/L ≈ 319,850 mg/L) is was overwhelmingly higher than the solubility of methylene blue dye. In essence, the standard state C° must be 1 mol/L as defined by IUPAC [40, 42, 45]. Therefore, scholars cannot freely select the value for the standard state C°.Table 2
Thermodynamic parameters for diclofenac adsorption onto F400 activated carbon [72].
T(K)Selected standard states (C° =1 mg/L?)Selected standard states (C° =1 mol/L)KL (L/mg)∆G°∆H°∆S°KL (L/mol)∆G°∆H°∆S°2880.0875.85–30.85–127.525765–24.33–30.85–22.752980.0477.58(y =176.8x – 0.532)13919–23.65(y = 3710.6x –2.7361)3080.0507.68(r2 =0.8629)14807–24.60(r2 =0.8558)3180.02210.106515–23.23Note: The unit of ∆G° (kJ/mol), ∆H°(kJ/mol), and ∆S° [J/(mol‧K)]; the information in parenthesis indicating the van’t Hoff equation.Table 3
Thermodynamic parameters for adsorbing phenol onto bispyridinium dibromides-modified montmorillonites.
T (K)KLC°KEqoThermodynamic parametersRemarks∆G°(kJ/mol)∆H°(kJ/mol)∆S°[J/(mol‧K)]1. Case 1 (the units of KL and C° are L/mg and mg/L)Modifiedand reported by [4]2980.0156 L/mg1 mg/L0.015610.31–14.48–83.213130.0116 L/mg1 mg/L0.011611.593280.0091 L/mg1 mg/L0.009112.802. Case 2 (the units of KL and C° are L/mmol and mmol/L)[4]2981.467 L/mmol1 mmol/L1.467–0.95–14.48–45.433131.096 L/mmol1 mmol/L1.096–0.243280.860 L/mmol1 mmol/L0.860.413. Case 3 (the units of KL and C° are L/g and g/L)[4]29815.59 L/g1 g/L15.59–6.81–14.48–25.7831311.65 L/g1 g/L11.65–6.393289.14 L/g1 g/L9.14–6.034. Case 4 (the units of KL and C° are L/mol and mol/L)Originally published by [68]2981467 L/mol1 mol/L1467–18.06–14.4812.013131096 L/mol1 mol/L1096–18.22328860 L/mol1 mol/L860–18.43To sum up, the standard thermodynamic equilibrium constant (KEqo) is always dimensionless (without any unit). This conclusion has been defined by the International Union of Pure and Applied Chemistry [39]. In fact, KEqois dimensionless because the activity (a) in Equation (24) is dimensionless. From Equation (27), the units of three parameters ΔG°, R, and T, respectively) are kJ/mol, kJ/(mol × K), and K; therefore, lnKEqo (or KEqo) must be dimensionless. Those discussions have been also stated by many scholars [21, 33, 52, 74, 75]. Adsorption processes in solid–liquid phases (i.e., the adsorption of phenol by active carbon) actually contain at least three important components: adsorbent (active carbon that is solid), adsorbate (phenol that is solute), and water (solvent). KEqo should be only dependent on temperatures, but KL is dependent on the concentrations of adsorbate. In addition, none of all experimental conditions of adsorption are conducted under strict standard conditions. Therefore, it is not easy to accurately determine the so-called “standard” thermodynamic parameters in liquid–phase adsorption systems. The so-called standard thermodynamic equilibrium constant might be obtained by considering some assumptions such as the standard references for pure solid and pure liquid phases being 1 mol/kg and 1 mol/L. Under such assumptions, the relationship between KEqo and KL can be expressed as Equation (36).


## 3. Some Mistakes Regarding the Application of the Langmuir Constant for Calculating the Thermodynamic Parameters

The first problem involves the standard state for adsorbate concentration (C°) in Equation (36). Some researchers directly applied KL (L/mg) as KEqofor calculating the thermodynamic parameters. In this case, the C° is tacitly selected as 1 mg/L [72]. This problem of this selection on the magnitudes (both sign and value) of thermodynamic parameters (i.e., Table 3) has been discussed in Section 2 and in the literature [4, 22, 23, 33].Recently, Najaflou et al. [76] applied the Langmuir constant for calculating the thermodynamic parameters of Pb adsorption by cellulose sulfate/chitosan aerogel. To convert KL from a dimensional constant to dimensionless one, they introduced a new conversation (Equation (38)). Clearly, this is a basic mistake. The authors understanded incorrectly the unit of KL that is liter per milligram of adsorbent (solid). The correction must be liter per milligram of adsorbate (solute). The unit of symbol m (defined in Equation (1)) is milligram of adsorbent (solid). The mass of adsorbate (in milligram) is remarkably different to that of adsorbent.
(38)KEqo=KLLmg×mVmgL?Reddy et al. [77] defined the equilibrium constant K (as Equation (39)) and applied it as KEqo for calculating the thermodynamic parameters of the adsorption of potential toxic metals ions onto the modified-biosorbent (derived from Moringa oleifera leaves). They expected that the constant K has a unit of L/g. Its unit is similar to that of the distribution coefficient KD or the Henry constant. Therefore, the thermodynamic parameters (Table 4) calculated based on this constant K do not bring physical meanings compared to those on the Langmuir constant KL (Table 4).
(39)KL/g=QmaxmggKLLmgTable 4
Thermodynamic parameters for adsorbing potential toxic metals ions by the modified-biosorbent.
As reported by Reddy et al. [77]As re-calculated by the authorQm(mg/g)KL(L/mg)K(L/g)∆G°∆H°∆S°KL(L/mol)KEqo∆G°∆H°∆S°1. For adsorbing Cd(II)293 K159.80.0274.317–3.5614.6161.8930353035–19.5311.96107.4303 K166.20.034.987–4.05(r2 =0.9740)33723372–20.47(r2 =0.9574)313 K171.40.0376.341–4.8141594159–21.692. For adsorbing Cu(II)293 K146.90.0243.525–3.0712.3352.6115251525–17.867.2685.88303 K151.30.0284.236–3.64(r2 =0.9964)17791779–18.85(r2 =0.8954)313 K167.90.0294.869–4.1218431843–19.573. For adsorbing Ni(II)293 K138.00.0212.899–2.5910.0042.9412321232–17.343.4771.01303 K148.80.0223.273–2.99(r2 =0.9961)12911291–18.05(r2 =0.999)313 K163.90.0233.769–3.4513501350–18.76Note: The unit of ∆G° (kJ/mol), ∆H° (kJ/mol), and ∆S° [J/(mol‧K)].


## 4. Derivation of the Distribution Coefficient and whether It Is Suitable for Applying the Calculation of Thermodynamic Parameters

Fifty years ago, Biggar and Cheung [78] developed the simple method for estimating the thermodynamic parameters of picloram adsorption by different soils. This method was later modified by Khan and Singh [79] for calculating the thermodynamic parameters of carbofuran adsorption by Sn(IV) arsenosilicate cation exchanger. The method modified by Khan and Singh [79] has been intensively applied in the literature later for calculating the thermodynamic parameters [9–12, 80]. Many researchers directly applied the distribution coefficient (KD; Equation (40)) as the standard thermodynamic equilibrium constant (KEqo) for calculating the thermodynamic parameters [7, 9, 10, 12]. In contrast, some others had converted the unit of KD before it was applied for the calculation [80–82]. A critical question whether both cases (direct and indirect) are appropriate for obtaining the thermodynamic parameters consistent with physical meanings is also discussed in this paper.According to Khan and Singh [79], the KD value was obtained by plotting ln(qe/Ce) versus qe and extrapolating to zero qe. The qe and Ce parameters are defined in Equation (1). After that, a straight-line fitted into the experimental data (with a high r2 value), and the intersection of the straight-line with the vertical axis provided the value of KD. Clearly, the KD value must be obtained from the different adsorbate concentrations of an adsorption isotherm [7]. The distribution coefficient KD obtained from this method still has a certain unit such as L/g or L/kg as described in Equation (40) or Equation (41), respectively.
(40)KD=qeCe=mg/gmg/L=Lgofadsorbentthatissolid(41)KD=qeCe=mol/kgmol/L=Lkgofadsorbentthatissolidwhere Ce and qe is defined in Equation (1), and their units—Ce (mg/L) and qe (mg/g)—are selected in this section because they are the most common use in the literature.A typical example expressing the idea of Khan and Singh [79] is provided by taking the full adsorption isotherm of methylene green dye by commercial activated carbon [24]. The full adsorption isotherm (the points of the equilibrium adsorption data: n = 19) is expressed in Figure 2(a). Based on the idea of Khan and Singh [79], the author continuously divided the full adsorption isotherm into two regions that (1) one is the adsorption isotherm containing outer points (n = 7) and (2) another is the adsorption isotherm obtained by removing the outers (n = 12). The values KD (linear method) and KL (non-linear method) calculated based on the three regions are provided in Table 5. Table 6 indicates the thermodynamic parameters of the adsorption process calculated based on KD and KL. Although the sign of ∆G°, ∆H°, and ∆S° calculated based on KD are relatively similar that based on KL, their magnitudes based on KD and KL are different. An important question is how to define the standard state (i.e., C°) for KD in Equation (40) or (41). This is because ∆G°, ∆H°, and ∆S° are defined as the standard thermodynamic parameters for an adsorption process. Notably, γAdsorbate cannot be ignored in several adsorption cases (i.e., Table 1), but it is not defined in two equations: Equations (40) and (41).Figure 2
(a) Adsorption isotherm of methylene green dye onto commercial activated carbon [24]; and (b)–(d) the different plots of qe/Cevs. qe for determining constant KD.Table 5
The calculating results for the constantsKD and KL based on Figure 2.
T(K)Whenqe (mg/g) vs. Ce (mg/L)Whenqe (mol/kg) vs. Ce (mol/L)KD (L/g)r2KL (L/mg)adj-R2KD (L/kg)r2KL (L/mol)adj-R21. Full adsorption isotherm (n =19)28871.500.28860.04400.9869715000.2886160680.986930385.120.35110.05880.9901851210.3511214350.990131892.930.31370.07060.9899929260.3137257510.98992. The outer region (n = 7)28897.790.10350.54270.8392977940.10351979720.8391303117.00.15810.51770.86351169730.15811888920.8635318123.90.10430.32700.86681238970.10431193180.86683. After removing the outer region (n =12) used for calculating the thermodynamic parameters2887.3750.98710.0390.996173750.9871141300.996130312.170.97650.0530.9952121010.9765194940.995231814.500.98460.0590.9977145020.9846216310.9977Table 6
Thermodynamic parameters of the dye adsorption process (Figure2) calculated based on KD and KL.
T(K)Whenqe (mg/g) vs. Ce (mg/L)Whenqe (mol/kg) vs. Ce (mol/L)KD(L/g)KL(L/mg)∆G°∆H°∆S°KD(L/kg)KL(L/mol)∆G°∆H°∆S°1. Calculated based onKD2887.375—–4.78517.2977.027375—–21.3317.29134.430312.17—–6.295(r2 =0.9425)12101—–23.68(r2 =0.9464)31814.50—–7.07114502—–25.332. Calculated based onKL288—0.039–22.8810.89117.5—14130–22.8810.89117.5303—0.053–24.88(r2 =0.9347)—19494–24.89(r2 =0.9346)318—0.059–26.39—21631–26.39Note: The unit of ∆G° (kJ/mol), ∆H°(kJ/mol), and ∆S° [J/(mol‧K)]; the information in parenthesis indicating the van’t Hoff equation; assuming KD as KEqo by Khan and Singh [79]; KL (L/mol) being considered as KEqo (when C° =1 mol/L).However, many researchers commonly do the adsorption study at only one initial adsorbate concentration (Co) and directly applied the qe and Ce values (examined at different temperatures) for calculating KD values [9, 10]. This is misconception from the method proposed by Khan and Singh [79]. Based on the raw data in Figure 2 [24], Table 7 shows the KD values and the thermodynamic parameters were calculated at each adsorbate concentration under different temperatures. At one initial adsorbate concentration, after equilibrium adsorption, Ce and qe values are obtained. The KD value is differently calculated based on those Ce and qe. The thermodynamic parameters obtained based on the KD (in this table) do not bring any physical meanings.Table 7
Thermodynamic parameters of the dye adsorption process (Figure2) calculated based on KD obtained at each initial adsorbate concentration under different temperatures.
Co(mg/L)Whenqe (mg/g) vs. Ce (mg/L)Whenqe (mol/kg) vs. Ce (mol/L)KD (L/g)∆G°∆H°∆S°r2KD (L/kg)∆G°∆H°∆S°r2288 K303 K318 K288 K303 K318 K288 K303 K318 K288 K303 K318 KLowest6.596.836.83–4.52–4.84–5.080.9018.850.774659168246824–21.06–22.24–23.340.9076.270.77450.6492.5573.71–9.40–11.41–11.379.8767.960.408506409253573700–25.94–28.81–29.639.87125.40.408112.5132.3149.1–11.31–12.31–13.237.1764.190.996112471132354149095–27.85–29.71–31.507.17121.60.996239.7249.6299.8–13.12–13.91–15.085.6164.840.861239742249593299814–29.66–31.31–33.345.61122.30.86129.6735.4036.02–8.12–8.99–9.484.9845.690.838296743540236020–24.66–26.39–27.744.98103.10.83812.2716.7824.94–6.00–7.10–8.5017.9683.040.991122671677824936–22.54–24.51–26.7717.96140.50.9916.708.4717.18–4.55–5.38–7.5223.7097.410.9076695847017177–21.09–22.79–25.7823.70154.90.9074.758.139.93–3.73–5.28–6.0718.8578.800.948474881329930–20.27–22.68–24.3318.85136.20.9483.275.026.31–2.83–4.06–4.8716.7768.290.979326650166306–19.38–21.47–23.1316.77125.70.9792.613.434.81–2.30–3.11–4.1515.4161.370.992261534314806–18.84–20.51–22.4215.41118.80.9922.122.553.33–1.80–2.36–3.1811.4245.780.982212125513332–18.34–19.76–21.4511.42103.20.9821.511.882.20–0.99–1.58–2.089.5336.570.997151218762199–17.53–18.99–20.359.5394.010.9971.151.391.66–0.34–0.83–1.359.3633.640.999115013891663–16.88–18.23–19.619.3691.080.9990.861.081.330.35–0.20–0.7510.9836.890.99986210821329–16.19–17.60–19.0210.9894.320.9990.530.640.791.531.140.6110.3930.670.994527637795–15.01–16.26–17.6610.3988.110.9940.390.470.522.231.931.737.0616.830.993396465521–14.32–15.47–16.547.0274.130.9950.320.380.422.742.462.277.2515.700.997319375426–13.80–14.93–16.017.3773.550.9980.270.310.363.182.972.737.4514.820.999265308355–13.36–14.44–15.537.4172.130.999Highest0.210.240.273.783.613.496.579.730.998206239268–12.76–13.79–14.786.6567.420.998Some summariesAverage25.0629.8733.66–2.91–3.64–4.3110.5946.90250562986633658–19.45–21.05–22.5710.60104.4Minimum0.210.240.27–13.12–13.91–15.080.909.730.408206239268–29.66–31.31–33.340.9067.420.408Maximum239.7249.6299.83.783.613.4923.7097.410.999239742249593299814–12.76–13.79–14.7823.70154.90.999Median2.613.434.81–2.30–3.11–4.159.5345.69261534314806–18.84–20.51–22.429.53103.1In general, Equation (40) can be expressed as the one-parametric Henry model (Equation (42)). In this case, the Henry constant (KH; L/g) is equal to the distribution coefficient (KD; L/g). In essence, the Henry model is a specific case of the Freundlich model (Equation (43)) when the exponent n of the Freundlich model is unity. Moreover, when the concentration of adsorbate in solution (~infinite dilution) is extremely low (this means KL and Ce << 1.0), the Langmuir model (Equation (44)) reduces to the linear Henry model (Equation (45)). Therefore, the constant KD might be obtained when the concentrations of adsorbate in solution (Ce) after adsorption are very low [5, 7]. However, under this situation, the adsorption sites available in adsorbent are not fully covered by adsorbate. This means that adsorption process does not reach equilibrium in two phases (solid and liquid). Figure 3 that shows the different shapes of adsorption isotherm in solid–liquid phases adapted from the reference [83]. The constant KH or KD is identified in an un-saturated adsorption region in Figure 3. Under this region, the adsorbing sites available in adsorbent are not fully covered (or occupied) by adsorbate.
(42)qe=KHCeHenrymodel(43)qe=KFCenFreundlichmodel(44)qe=QmaxKLCe1+KLCeLangmuirmodel(45)qe=QmaxKLCe=KHCewithKH=Qmax×KL(46)KD=qeCe=QmaxKLobtainedfromEquation(44),whenCe⟶0Figure 3
Different shapes of the adsorption isotherms classified by Moreno-Castilla [83].whereKH (L/g) is the Henry constant; Qmax (mg/g) is the Langmuir maximum adsorption capacity of an adsorbent under optimal operation conditions (pH, contact time, etc.) at a constant temperature; KL (L/mg) is the Langmuir equilibrium constant; KF is the Freundlich constant (mg/g)/(mg/L)n, and n is the exponent of the Freundlich model (0 < n < 1).As shown in Equation (40), KD is a dimensional coefficient, with a common unit being L/g. Therefore, it cannot be directly applied as theconstant KEqo for calculating the thermodynamic parameters. This problem has been intensively discussed in the literature [2, 7, 18–21, 84–86].Some authors recommended to convert the unit ofKD (L/g) into KEqo (no unit) by simply multiplying KD value by 1000 (Equation (47)) [82, 87]. The author considered the density of pure water (ρwater) is ~1.0 g/mL (equal to 1000 g/L). Such idea has been followed by some other researchers [70, 71, 75, 81]. Unfortunately, this suggestion is not correct, and the mistake has been thoroughly discussed elsewhere [2, 5, 58]. In fact, as shown in Equation (40), the unit of KD is liters of solution per grams of adsorbent (not adsorbate!), so Equation (47) is misconception. This problem has been early identified and discussed by Liu [1].
(47)KEqo=KD×ρwater=Lgofadsorbent×1000gofwaterLSeveral other authors tried to convert the unit ofKD (L/g) intoKEqo (no unit) using Equation (48) or (49) [11, 12, 88]. A critical comment on this conversion method has been published by some authors [7, 89]. Similarly, Lima and co-workers [2] suggested that the KD values obtained from Equation (48) or (49) cannot be applied as KEqofor calculating the thermodynamic parameters.
(48)KEqo=KD×mV=qeCe×mV=Lgofadsorbent×gofadsorbentLofadsorbate(49)KEqo=KD×mV=qeCe×mV=Co−CeVCe×m×mV=Co−CeCeFurthermore, Zhou [80] recommended to apply Equation (50) and Equation (51) to convert the unit of KD (L/g) into KEqo (no unit). However, those two methods have not received the concerns and agreements of the communities in the field. The main reason of the disagreement has been thoroughly analysed by Rahmani-Sani and colleagues [90]. This is also main reason why the distribution coefficient KD (not true adsorption equilibrium constant) cannot be used as the standard thermodynamic equilibrium constantKEqofor calculating the thermodynamic parameters. The thermodynamic parameters that are calculated based on the distribution coefficient KD do not bring physical meanings. An analogous conclusion has been claimed by some other scholars [1, 2, 5, 20, 21].
(50)KD=Γ×aSACe=kg/m2×1/mkg/m3=kg/m3ofsolid?kg/m3ofliquid?(51)KD=Γtthickness×Ce=kg/m2ofsolid?m×kg/m3ofliquid?where Г (kg/m2) is surface concentration; aSA (1/m) is the surface area (m2) available for adsorption per unit volume (m3); Ce (mg/L or kg/m3) is the equilibrium concentration of adsorbate; and tthickness (m) is the thickness of the by Zhou adsorbed layer. The information was reported as reference [80].On the basis of the Henry model (Equation (42)), Sawafta and Shahwan [91] proposed a method to estimate thermodynamic equilibrium constant of the adsorption process of methylene blue in water and water–ethanol solution by iron nanoparticles. The authors rearranged Equation (1) to obtainCe as Equation (52), and the Henry model (Equation (42)) will become Equation (53). After rearranging, Equation (54) is obtained. The parameters (qe, Co, m, and V) in Equation (54) have been defined in Equation (1). They reported that the KH constant as obtained from the plot of qevs. Co (in Equation (54) that is a linear form as y = ax + b) is dimensionless and equal to thermodynamic equilibrium constant. The KH value is determined from the slope of Equation (54). A critical question is what the meanings of the y-intercept of this linear equation is. In addition, when the unit of qe (mg/g) and Co (mg/L) is introduced in Equation (54), Equation (55) is obtained. Cleary, the unit of KH (Equation (55)) still be L/g that is similar to KD (L/g) as early discussed. Therefore, the method that was recently proposed by Sawafta and Shahwan [91] for determining the constant (that is similar to KD; so it cannot be equal to so-called thermodynamic equilibrium constant) is not suitable for calculating the thermodynamic parameters.
(52)Ce=Co–qeVm(53)qe=KH×Ce=KH×Co–qeVm(54)qe=KH1+KH/VmCo(55)mgg=KH1+KH×gofadsorbent/L×mggRecently, Chen et al. [92] gave a new definition for the distribution coefficient. This new coefficient is named as “the standard equilibrium constant” KDo. The relationship between KD and KDo is shown in Equation (56). The authors [92] defined two parameters (q° and C°, respectively) as the standard states of solution in the solid and liquid phases. The authors assumed that q° and C° are 1 mol/kg and 1 mol/L. They concluded that KDois calculated from Equation (56) is dimensionless. Unfortunately, the definition of the standard equilibrium constant KDo is incorrect from beginning. Lima et al. [21] commented that the term of q° in Equation (56) does not exist in the science and the literature. Form Equation (56), Lima et al. [21] estimated that q° is obtained from Equation (57) although Chen et al. [92] did not provide this equation. If Coo and Ceo are 1 mol/L, q° will be zero (i.e., 0 mol/kg). This means that KDo is a value undetermined or tending to the infinite according to the concept of functional limit [21].
(56)KDo=qe/q°Ce/C°=KD×C°q°(57)q°=Coo–Ceom°V°where C° and q° that are defined by Chen et al. [92] are the standard states of solution in liquid and solid, respectively.To sum up, the (direct and indirect) applications of the distribution coefficient (KD; Equation (40) or (41)) as the standard thermodynamic equilibrium constant (KEqo; Equation (26)) for calculating the thermodynamic parameters of solid–liquid adsorption process are not feasible and thus should be avoided because of the unit problem and other aforementioned reasons (i.e., it is not true adsorption equilibrium constant). The equilibrium constant of some adsorption models (i.e., KL from the Langmuir model) can be applied as the thermodynamic equilibrium constant by appropriately considering [Adsorbate]° and γAdsorbate (Equation (26))).


## 5. Feasibility of Applying the Freundlich Constant (KF) for Calculating Thermodynamic Parameters?

Unlike, the Langmuir model, the Freundlich model (Equation (43)) is an earliest empirical equation without physical meanings [93]. From Equation (43), the unit of the Freundlich constant KF that is (mol/kg)/(mol/L)n is correctly obtained in Equation (58). Some authors made a mistake in presenting the unit of KF as L/mol (or L/mg) or even mol/g (or mg/g [77, 85]) without considering the magnitude of its exponent. This mistake has been discussed elsewhere [94, 95]. Only the case of its exponent n = 1 (a linear isotherm), the unit of KF reduces from (mol/kg)/(mol/L)n to L/kg and KF will be equal to KH. For example, when n = 1, Chao et el. [96] reported the KF values (1023, 442, 252, 199, 120 L/kg) for adsorbing organic compounds (propylbenzene, ethylbenzene, toluene, benzene, trichloromethane) by titanate nanotubes modified with octadecyltrichlorosilane were inversely proportional to their water solubilities (55, 152, 515, 1780, 7900 mg/L, respectively).
(58)KF=qeCen=mol/kgmol/LnHúmpola and co-workers [69] applied the Freundlich constant KF as the constant KEqo for calculating the thermodynamic parameters of the adsorption process of phenol onto different kinds of activated carbon. However, they reported the incorrect unit as [(mol(1–n) × Ln)/g]. Although the authors reported the thermodynamic equilibrium constant in Equation (27) was L/mol, they did not provide detail information how to convert the unit of KF from [(mol(1–n) × Ln)/g] to L/mol. A similar mis-application of KF for calculating the thermodynamic parameters of the adsorption of Co2+ ion by chitosan-aluminium oxide composite was reported by Ma et al. [17].Salunkhe [97] applied the constant KF for calculating the thermodynamic parameters of the adsorption process of methylene blue by hydrogels. The authors have a mistake regarding the unit of KF (L/mg). They converted the unit KF from L/mg to L/mol and then applied it as KEqo for calculating the thermodynamic parameters (Table 8). The correct thermodynamic parameters for this process should be calculated based on KL (Table 8).Table 8
Thermodynamic parameters of the adsorption process of methylene blue onto hydrogels.
T(K)KF∗(L/mg)?KL∗(L/mg)KEqovan’t Hoff equationThermodynamic parameters∆G°(kJ/mol)∆H°(kJ/mol)∆S°[J/(mol × K)]1. Based on the Freundlich constant (incorrect)29512.38∗—3939592y = –1188x+ 19.19(r2 =0.9191)–37.25∗9.87159.630513.17∗—4212840–38.68∗(0.14∗)?(2.3∗)?31515.93∗—5094315–40.45∗2. Directly based on the Freundlich constant (incorrect)29512.38∗—12.38∗y = –1188x+ 6.52(r2 =0.9191)–6.169.8754.1730513.17∗—13.17∗–6.5431515.93∗—15.93∗–7.253. Directly based on the Langmuir constant (incorrect)295—0.00392∗0.00392y = –2692x+ 3.53(r2 =0.9139)13.5922.3829.37305—0.00454∗0.0045413.68315—0.00702∗0.0070212.994. Based on the Langmuir constant (correct)295—0.00392∗1254y = –2691.6x+ 16.21(r2 =0.9139)–17.5022.38134.8305—0.00454∗1452–18.46315—0.00702∗2245–20.21Note:∗Results are reported by the author [97]. The exponent n is 0.9843 at 295 K, 1.0095 at 305 K, and 1.0572 at 315 K [97].In 2015, Ghosal and Gupta [16] introduced the relationship between KEqo and KF (mg/g)/(mg/L)n as Equation (59) by considering the mass of water per litre (actually, it is the density of pure water ρwater; g/L). This might be the first paper on this unit conversion because they did not give any reference for this idea and equation. The idea of using ρwater to convert KF from dimensional to dimensionless based on the previous publication of Milonjić [87]. The authors [16] reported that the KEqo constant obtained from Equation (59) is dimensionless. Many authors have applied Equation (59) for calculating the thermodynamic parameters [14, 15, 98, 99].It is necessary to verify whether theKEqo constant calculated from Equation (59) is actually dimensionless. The analysis results (Equations (60)–(63)) demonstrated that the information in Equation (59) is completely error. This is because of the significant difference among the mass of adsorbent (solid), adsorbate (solute), and water (solvent). This problem has been identified by other authors [65, 100]. Therefore, the thermodynamic parameters calculated based on the Freundlich constant are not true and do not have physical meanings [24, 65, 100].
(59)KEqo=KF×ρwater×1ρwater/10001–n(60)KEqo=mgofadsorbate/gofadsorbentmgofadsorbate/Ln×gofwaterL×1gofwater/L×10001–n(61)KEqo=mgofadsorbategofadsorbent×Lmgofadsorbaten×gofwaterL×Lmgofwater1–n(62)KEqo=mgofadsorbategofadsorbent×gofwaterL×Lmgofadsorbaten×L/mgofwaterL/mgofwatern(63)KEqo=mgofadsorbategofadsorbent×gofwaterL×Lmgofwater×LmgofadsorbatenmgofwaterLnLin et al. [101] applied Equations (64)–(66) for calculating the thermodynamic parameters of the adsorption process of phenol by hydroxyapatite solid. They calculated ∆H° based on the plot of ln(1/Ce) vs. 1/T in Equation (64). The unit of KF reported by Lin et al. [101] is L/mg. ∆G° was directly calculated based on the exponent of the Freundlich model. Their result in Table 9 is doubtfully.
(64)ln1Ce=lnKF–∆H°RT(65)∆G°=–nRT(66)∆S°=∆H°–∆G°TTable 9
Thermodynamic parameters of the adsorption process of phenol onto hydroxyapatite.
T(K)KL(L/mg)KEqoThermodynamic parameters∆G°(kJ/mol)∆H°(kJ/mol)∆S°[J/(mol × K)]1. As reported by Lin et al. [101] using Equations (64)–(66)293——−2.036.4829.0313——−2.2627.9333——−2.3226.42. As recalculated based onKL (suggested)2930.1225∗12145−22.9112.34120.03130.1482∗14693−24.973330.2264∗22445−27.74Note:∗Values are reported by the author [101].


## 6. Considering Other Adsorption Equilibrium Constants Related to Adsorption Isotherms

Apart from the constantKL (L/mol) of the Langmuir model that can been applied for calculating the thermodynamic parameters of various adsorption processes, several other constants from the some adsorption isotherm models have been also considered as KEqo for calculating the thermodynamic parameters [21, 63].Recently, Lima et al. [102] initially gave some new ideas on applying other constants of the adsorption isotherms for calculating the thermodynamic parameters of adsorption process in solid–liquid phases. The most important consideration is the unit of those constants that must be expressed as L/mol (as the constant KL). In this case, those constants are defined as the adsorption equilibrium constants, and they can be used to replace KL in Equation (36) for calculating the thermodynamic parameters. From those ideas, Tran et al. [24] successfully applied the constants of various adsorption models (i.e., the Langmuir, Liu, Sips, Toth, Khan, Hill, Redlich–Peterson, Radke–Prausnitz, and Koble–Corrigan models) for calculating the thermodynamic parameters of adsorbing methylene green (MG) dye by commercial activated carbon (CAC) (Table 10). Those models possess the features of the Freundlich and Langmuir models.Table 10
The thermodynamic parameters of the adsorption process of methylene green dye using commercial activated carbon calculated based on various adsorption equilibrium constants of the isotherm models [24].
T(K)KModelnmodelKEqoThermodynamic parameters∆G°Linear methodNon-linear method∆H°∆S°∆H°∆S°1. Langmuir model:KModel = KL (L/mol)28816,068—16,068–23.1912.01122.311.68121.2430321,435—21,435–25.1231825,752—25,752–26.852. Langmuir–Freundlich model:KModel = KLF (L/mol)28813,7340.743013,734–22.8111.73120.1118.870.972930318,3130.755818,313–24.7331821,7670.739421,767–26.413. Liu model:KModel = KLiu (mol/L)28872.81 × 10–60.742913,735–22.8111.72120.1118.860.972630354.59 × 10–60.755818,318–24.7331845.94 × 10–60.739421,767–26.414. Sips model:KModel = KSipL/mol1/nSips28811920.743513,815–22.8011.71120.011.38118.9530316710.756118,331–24.7131816280.740321,881–26.425. Khan model:KModel = KKhan (L/mol)28821,5790.957921,579–23.9016.73141.116.51140.0130331,0700.950331,070–26.0631841,6880.942941,688–28.13Note: The unit of ∆G°, ∆H° and ∆S° was kJ/mol and J/(mol × K), respectively.Among those models, the Langmuir–Freundlich model (or the Liu model) have been intensively applied for calculating the thermodynamic parameters. It is hard to verify the original paper for the Langmuir–Freundlich model. In the literature, this model is often expressed in different forms. However, the most common form is expressed as Equation (67). Some authors found that this form (Equation (67)) is similar to the Liu model (Equation (68)) [24, 102]. The relationship between two models is expressed in Equation (70). When summiting Equation (69) into Equation (67), the Langmuir–Freundlich model will become the Liu model. Therefore, it can be concluded that the Liu model and Langmuir–Freundlich model are the same fundamentals and derivations.
(67)qe=QLFKLFCenLF1+KLFCenLF(68)qe=QLiuCenLiuKLiunLiu+CenLiu(69)KLF=1KLiu(70)qe=QLF1/KLiunLFCenLF1+1/KLiunLFCenLF=QLFCenLFKLiunLF+CenLF(71)KLiueq=1KLiuwhere QLiu = QLF (mol/kg) is the MAC of the models; KLiu (mol/L) is the Liu constant; KLiu(eq) (obtained from Equation (71)) = KLF (L/mol) is the adsorption equilibrium constant of the Liu and Langmuir–Freundlich models; and nLF (dimensionless) = nLiu is the exponents of the Langmuir–Freundlich and Liu models.In 2003, Liu and coworkers [103] initially developed a general model derived from a thermodynamic approach (Equation (68)). The equilibrium constant KLiu(eq) of this model is appropriate for calculating the thermodynamic parameters, and so it is then applied by many scholars [24, 104, 105]. Surprisingly, the authors reported the constant KLiu without any unit [103, 106]. However, as analyzed in Equation (72), the unit of the constant KLiu must be mol/L (when considering qe = mol/kg and Ce = mol/L). This is consistent with its unit expressed in Equation (69). The authors [103, 106] did not provide a strict limitation for the exponent nLiu of this model. They reported the nLiu value of 0.52 for adsorbing Cu2+, 1.02 for adsorbing Zn2+, and 1.2 for adsorbing Cd2+ by aerobic granules [103]. However, Lima et al. [102] suggested that this exponent must be a positive value (nLiu> 0). The relationship between KEqo and KLiu(eq) is expressed as Equation (73) [24, 102]. Some researchers have applied KLiu(eq) as KEqo for calculating the thermodynamic parameters of some adsorption processes (Table 10) [24, 104].
(72)molkg=mol/kg×mol/LnLiuKLiunLiu+mol/LnLiu(73)KEqo=KLiueqγAdsorbateCoBecause the Langmuir–Freundlich model has the same form to the Liu model, the name of two models should be integrated and called as the Liu model. The name of the Langmuir–Freundlich model should be considered as the general adsorption isotherm models combining the characteristic of the Langmuir model and Freundlich model. The isotherm models (Sips, Khan, etc.) that can reduce to the Langmuir or the Freundlich model under specific conditions are called as the Langmuir–Freundlich typed model.Another feasible consideration is the Sips model [107]. In nature, it is one of the typical Langmuir model-typed equation. Assuming one adsorbate is occupied by the adsorbing sites n of adsorbent. On the average, the Sips model for adsorbing under liquid–solid phases is given as Equation (74). Lima et al. [102] suggested that the exponent of this model should be in a strict range (0 < 1/nSips ≤ 1). When nSips = 1, this model reduces to the Langmuir model. The adsorption equilibrium constant of this model KSips(eq) (L/mol) is defined as Equation (75) that can be used as KEqo (Equation (76)) for calculating the thermodynamic parameters (Table 10) [24, 102].
(74)qe=QSipsKSipsCe1/nSips1+KSipsCe1/nSips(75)KSipseq=KSips1/nSips(76)KEqo=KSipseqγAdsorbateCowhere QSips is the MAC of Sips model (mol/kg); KSips is its constant L/mol1/nSips; nSips is its exponent (dimensionless).Similar to the Langmuir models, the unit of the AEC of Khan model (KKhan) is L/mol. However, the constant KKhan is independent on its exponent. The exponent nKhanshould be higher than zero Therefore, it is possible to directly apply it as KEqo for calculating the thermodynamic parameters (Table 10) [24, 102]. In 1997, Khan et al. [25] modified their previous model (proposed for describing the adsorption data of multi-components) and then applied it for describing the adsorption behavior of phenol, p-nitrophenol, or p-chlorophenol in single solution by activated carbon. This modified model is commonly known as the Khan model (Equation (77)). When nKhan = 1 this model is mathematically equivalent the Langmuir model. Khan et al. [25] reported that nKhan (in both their current study and extracted from the literature) for adsorbing various single organic adsorbates in water ranged from 0.55 to 5.84. However, it is necessary to set the limitation for this exponent.
(77)qe=QKhanKKhanCe1+KKhanCenKhanwhereQKhan is the MAC estimated by the Khan model (mol/kg); KKhan is the AEC of this model (L/mol); and nKhan is its exponent (dimensionless).


## 7. Some Suggestions for Improving the Quality of Papers Published in This Field

Science and technology are always changing. Some traditional methods or previous comments might be suitable and correct at the publication time. However, in the future, they might be not appropriate or even not correct.It is notable that none of all publications in high-impact journals are always correct. In essence, a manuscript submitted to the journal needs to undergo a reviewing process. The submitted work is often accepted for publication only based some peoples: reviewers and editors (but not all researchers in the field). Sometimes, some people who are named as experts are beginners (i.e., doctoral students); meanwhile, other people have never committed themselves to do experiments on adsorption or write the papers in this field.Therefore, the present author has some suggestions for beginners. When researchers (less experienced) start to do experiments and explain their data, they should wonder whether the methods, information, and results in the published papers or the books are appropriate for their current studies. In fact, beginners often follow or directly use the previously published methods of their groups/labs for their current works; for example, in the case of applying the unit conversion of the Freundlich constant [16]. As earlier discussed, none all publications prove correct information and accuracy method; therefore, open-minded beginners should read more and more papers (especially textbooks) published from different author groups and publishers. Importantly, researchers should give more updated citations when writing their manuscripts. This is an effective way to avoid the repeat of the similar mistakes discussed by other scholars in the literature [108].For example, in 2009, Shuibo and co-workers [109] directly applied the Langmuir constant KL (L/mg) for calculating the thermodynamic parameters of the process of uranium (VI) adsorption in water using the hematite-based adsorbent. In the same year, Milonjić [82] wrote a comment work on this direct application and suggested a conversion relation (Equation (78)). The author considered the density of pure water (ρwater) is ~1.0 g/mL (equal to 1,000,000 mg/L or 106 mg/L). This equation has been then applied by many scholars [98, 110–112]. However, ten years later, Zhou and Zhou [58] published a comment paper to analyse the feasibility of the previous comment made by Milonjić [82]. Zhou and Zhou [58] concluded that Equation (78) is certainly error because the difference between the mass of adsorbate and water (Equation (79)). Table 11 shows the comparison of the thermodynamic parameters that are obtained from (1) directly applying KL (L/mg) as KEqo [109] and (2) indirectly applying KL (L/mg) by the unit conversions [58, 82].
(78)KEqo=KL×106(79)KEqo=KL×106ρwater=Lmgofadsorbate×106mgofwatermLTable 11
Comparison of thermodynamic parameters for the adsorption process of U(VI) on hematite obtained from different calculation methods.
T(K)KEqo∆G°Linear method (Eq. (31))Non-linear method (Eq. (30))∆H°∆S°∆H°∆S°1.KL(L/mg) as the original report by Shuibo et al. [109]2931.01–0.02425.08117.255.29517.953051.03–0.07503181.19–0.45992.KL(L/mg) × 106 as the comment by Milonjić [82]2931,010,000–33.685.081132.115.295132.83051,030,000–35.113181,190,000–36.993.KL(L/mol) as the comment by Zhou and Zhou [58]293240,400–30.185.087120.25.301120.9305245,200–31.47318283,300–33.19Note: The unit of ∆G° (kJ/mol), ∆H° (kJ/mol), and ∆S° [J/(mol × K)]; and thermodynamic parameters recalculated based on the literature [58, 82, 109].Another example is the modified Langmuir model (Equation (80)) that was proposed by Azizian and co-workers [4]. As discussed in Section2, the constant KEqo is obtained by assuming that the standard state (or reference state) for pure liquids and pure solids is 1 mol/L and 1 mol/kg, respectively. However, some authors did not agree to this assumption [4, 5, 72]. Therefore, Azizian and co-workers [4] developed a revised form of the Langmuir isotherm using the aspect of adsorption and desorption kinetics. The authors concluded that the modified Langmuir constant (KML) in Equation (80) is a dimensionless quantity. Therefore, they suggested that KML can be directly applied for calculating the thermodynamic parameters. However, the theory on derivation of KML from thermodynamics aspects is missed. Different scholars made two comments on some problems of the modified Langmuir model [113, 114]. Unlike KL, the constant KML does not dependent on the units of qe and Ce used (Table 12) [24]. This constant (dimensionless) is a different principle to the adsorption equilibrium (dimensional such as L/mol) related to adsorption isotherm models such as the Langmuir, Lui, Sips, and Khan models. In addition, a question is how the authors can define activity coefficient for adsorbate (solute) in Equation (25). Therefore, the direct application of KML for calculating the thermodynamic parameters should be done with a thorough consideration because it might not be suitable for all cases of adsorption studies [24].
(80)qe=QMLKMLCeCs−Ce+KMLCewhere CS is the saturation concentration of solute in bulk phase that is previously defined in Equation (37); QML (mol/kg) is the modified Langmuir MAC of adsorbent; and KML (dimensionless) is the modified Langmuir constant.Table 12
Comparison of relevant parameters of the Langmuir and modified Langmuir models obtained from different units (qe and Ce) of the plots of MG adsorption isotherm by CAC [24].
Adsorption isothermqe (mg/g) vs.Ce (mg/L)qe (mmol/g) vs.Ce (mmol/L)qe (mol/kg) vs.Ce (mol/L)1. Langmuir modelQL211.2 mg/g0.578 mmol/g0.578 mol/kgKL0.059 L/mg21.44 L/mmol21435adj-R20.9900.9900.990red-χ269.45.2E-045.2E-042. Modified Langmuir modelQML210.6 mg/g0.577 mmol/g0.577 mol/kgKML588.4588.4588.4adj-R20.9900.9900.990red-χ269.45.2E-045.2E-04Some uncertain conclusions or questionable statements should be not continued to use in this field. For example, in 1998, Gupta [115] investigated the adsorption of nickel and copper ions using activated slag. The author concluded that “positive values of entropy (∆S°) reflect the affinity of the adsorbent material for the metal ions and sugges some structural changes in adsorbate and adsorbent”. The doubtful conclusion (positive values ∆S° suggest some structural changes in adsorbate and adsorbent) is drawn without supported by relevant experimental data (i.e., comparing the structure of adsorbent before and after adsorption). Other researchers have early commented on above conclusions; for example: “the positive entropy change during adsorption is the result of structural changes to the solute or adsorbent or the release of other ions by an ion exchange mechanism by the solute, without experimental or theoretical proof” [26]. However, many other researchers have cited the paper of Gupta for similar conclusions such as “the positive values of ΔS confirm a high preference of Methylene Blue molecules for the NLP surface and suggest possibility of some structural changes or readjustments in the Methylene Blue–NLP adsorption complex” [13]. Reddy et al. [77] concluded that “ the positive values of ΔS° show increased randomness at the solid/solution interface with some structural changes in the sorbate (hydrated metal ions) and biosorbent”.Another example is the information on the sign of ∆H°. In a review article, Hu et al. [116] concluded that “the value of KL should decrease with the increase in temperature since adsorption was usually an exothermic process (∆H° < 0)”. This conclusion is proposed without any experimental data supported. The value of KL can increase or decrease when solution temperatures increase [26]. For example, the constants KL of the adsorption process of phenol onto hydroxyapatite increased from 0.1225 L/mg to 0.2264 L/mg when the temperatures increased from 293 K to 333 K; as a result, the adsorption process was endothermic (∆H° > 0; +12.34 kJ/mol; Table 9) [101]. In contrast, the constants KL of the adsorption process of diclofenac adsorption onto F400 activated carbon decreased from 0.087 L/mg to 0.022 L/mg (∆H° < 0; –30.85 kJ/mol; Table 2) when the temperature increased from 288 K to 318 K [72].Lastly, Luo et al. [68] wrote “the ΔG° value is in the range of 0 to −20 kJ/mol and −80 to −400 kJ/mol for physical and chemical adsorptions, respectively. In this study, the ΔG° values are close to −18 kJ/mol, indicating that the adsorptions are mainly physical in nature”. Conclusions regarding physisorption or chemisorption must be proposed based on the magnitude of the standard change in enthalpy ΔH°, not standard Gibbs energy change ΔG°. In general, chemisorption (i.e., ΔH° >80 kJ/mol [104]) has a higher magnitude of ΔH° than physisorption. ΔH° value for covalent bonds (chemisorption) is often in the range 200–800 kJ/mol [117], while that for hydrogen bonds (physisorption) is lower than 40 kJ/mol [118].


## 8. Conclusions

The general form of standard dimensionless thermodynamic constantKEqo of the solid–liquid adsorption is fully expressed as Equation (25). Under some assumptions, Equation (25) decreases to Equation (26) and then commonly expressed as Equation (36): KEqo=KL/γAdsorbateCo. It is necessary to define or calculate two important parameters: γadsorabte (dimensionless) and C° (1 mol/L by definition). Adsorption equilibrium constants are applied in Equation (36) must be dimensional (their units must be L/mol like the Langmuir adsorption equilibrium constant). The plot of adsorption isotherm should be presented as qe (mol/kg) versus Ce (mol/L) in studying adsorption thermodynamics. The non-linear technique is always applied for calculating the relevant parameters of adsorption isotherm models. The adsorption equilibrium constants (i.e., KL, KLF = KLiu(eq), KSips(eq), and KKhan) of the Langmuir, Langmuir–Freundlich or Liu, Sips, and Khan models can be considered as KEqo and applied for calculating the thermodynamic parameters of adsorption process: so-called standard Gibbs energy change (∆G°; kJ/mol), standard enthalpy change (∆H°, kJ/mol); and standard entropy change [∆S°; J/(mol × K)]. The non-linear form of the van’t Hoff equation should be applied for calculating ∆H° and ∆S° to minimize some error functions. The negative value of ∆G° does not imply that adsorption processes occur spontaneously. The sign of ∆G° indicates the magnitude of KEqo. Mechanism involved in chemisorption is considered based on the magnitude of ∆H° (>80 kJ/mol), not ∆G°.The distribution constant (KD) and Freundlich constant (KF) are not a true adsorption equilibrium constant. They are not equal or converted toKEqo. Thermodynamic parameters calculated based on KD and KF do not bring physical meanings. The constant KD is often used of an analyte in two immiscible solvents such as water and n-octanol (hydrophobic). The application of KD or KF for calculating the thermodynamic parameters should be avoided. The author hopes that this paper will be helpful for other researchers in this filed.



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