TMMarcel Dekker, Inc. New York BaselADSORPTIONTheory, Modeling, and Analysisedited byJzsef TthUniversity of MiskolcMiskolc-Egyetemvros, HungaryCopyright 2001 by Marcel Dekker, Inc. All Rights Reserved.ISBN: 0-8247-0747-8This book is printed on acid-free paper.HeadquartersMarcel Dekker, Inc.270 Madison Avenue, New York, NY 10016tel: 212-696-9000; fax: 212-685-4540Eastern Hemisphere DistributionMarcel Dekker AGHutgasse 4, Postfach 812, CH-4001 Basel, Switzerlandtel: 41-61-261-8482; fax: 41-61-261-8896World Wide Webhttp:==www.dekker.comThe publisher offers discounts on this book when ordered in bulk quantities. For more infor-mation, write to Special Sales=Professional Marketing at the headquarters address above.Copyright # 2002 by Marcel Dekker, Inc. All Rights Reserved.Neither this book nor any part may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopying, microlming, and recording, orby any information storage and retrieval system, without permission in writing from thepublisher.Current printing (last digit):10 9 8 7 6 5 4 3 2 1PRINTED IN THE UNITED STATES OF AMERICAPrefaceThis book presents some apparent divergences, that is, its content branches off in manydirections. This fact is reected in the titles of the chapters and the methods applied indiscussing the problems of physical adsorption. It is not accidental. I aimed to prove that theproblems of physical adsorption, in spite of the ramied research elds, have similar or identicalroots. These statements mean that this book is 1) diverse, but still unied and 2) classical, butstill modern. The book contains monographs at a scientic level and some chapters include partsthat can be used by Ph.D.-level students or by researchers working in industry. Here are someexamples. According to the classical theories of adsorption (dynamic equilibrium or statisticalmechanics), the isotherm equations (Langmuir, Volmer, FowlerGuggenheim, deBoer, Hobson,Dubinin, etc.) and the corresponding thermodynamic functions of adsorption (entropy, enthalpy,free energy) include, in any form, the expression 1 Y, where Y is the coverage and, therefore,0 < Y < 1. This means that if the expression 1 Yappears as denominator in any of the above-mentioned relationships, then in the limiting caselimY=1(1 Y) = 0these functions tend to innity. Perhaps the oldest thermodynamical inconsistency appears inPolanyis equation, which expresses the adsorption potential with the following relationship:Pa = RT ln p0p where p0 is the saturation pressure. It is clear thatlimp0Pa = oThe mathematical and thermodynamical consequences of these facts are the following:1. The monolayer adsorption can be completed only when the equilibrium pressure isinnitely great.2. The change in thermodynamic functions are also innitely great when the monolayercapacity is completed.3. The adsorption potential tends to innity when p tends to zero.iiiAll consequences are physically and thermodynamically nonsense; however, in spite of this fact,the functions and isotherms having these contradictions can be applied excellently in practice.This statement is explicitly proven in Chapters 2, 3, 4, 6, 13, and 15, in which the authors applyLangmuirs and=or Polanyis equation to explain and describe the experimentally measured data.The reason for this is very simple: Because the measured data are far from the limiting cases(Y 1 or p 0), the deviations caused by the unreal values of thermodynamic functions arenot observable. This problem is worth mentioning because in all chapters of this bookexplicitly or implicitlythe question of thermodynamic consistency or inconsistency emerges,and the rst chapter tries to answer this question. However, independent of this problem, everychapter includes many new approaches to the topics discussed.The chapters can be divided into two parts: Chapters 19 deal mostly with gassolidadsorption and Chapters 1015 deal with liquidsolid adsorption. Chapter 2 discusses the gassolid adsorption on heterogeneous surfaces and provides an excellent and up-to-date overview ofthe recent literature, giving new results and aspects for a better and deeper understanding of theproblem in question. The same statements are valid for Chapters 47. In Chapters 8 and 9, theproblems of adsorption kinetics, using quite different methods, are discussed; however, thesemethods are successful from both a theoretical and a practical point of view. The liquidsolidadsorption discussed in Chapters 1015 can be regarded as developments and=or continuationsof Everetts and Shays work done in the 1960s and 1970s.In summary, I hope that this book gives a cross section of the recent theoretical andpractical results achieved in gassolid and liquidsolid adsorption, and it can be proved that themethods of discussion (modeling, analysis) have the same root. The interpretations can be tracedback to thermodynamically exact and consistent considerations.Jozsef To thiv PrefaceContentsPreface iiiContributors vii1. Uniform and Thermodynamically Consistent Interpretation of Adsorption Isotherms 1Jozsef Toth2. Adsorption on Heterogeneous Surfaces 105Malgorzata Borowko3. Models for the Pore-Size Distribution of Microporous Materials from a SingleAdsorption Isotherm 175Salil U. Rege and Ralph T. Yang4. Adsorption Isotherms for the Supercritical Region 211Li Zhou5. Irreversible Adsorption of Particles 251Zbigniew Adamczyk6. Multicomponent Adsorption: Principles and Models 375Alexander A. Shapiro and Erling H. Stenby7. Rare-Gas Adsorption 433Angel Mulero and Francisco Cuadros8. Ab Fine Problems in Physical Chemistry and the Analysis of AdsorptionDesorption Kinetics 509Gianfranco Cerofolini9. Stochastic Modeling of Adsorption Kinetics 537Seung-Mok Lee10. Adsorption from Liquid Mixtures on Solid Surfaces 573Imre Dekany and Ferenc Bergerv11. Surface Complexation Models of Adsorption: A Critical Survey in the Contextof Experimental Data 631Johannes Lu tzenkirchen12. Adsorption from Electrolyte Solutions 711Etelka Tombacz13. Polymer Adsorption at Solid Surfaces 743Vladimir Nikolajevich Kislenko14. Modeling of Protein Adsorption Equilibrium at Hydrophobic SolidWaterInterfaces 803Kamal Al-Malah15. Protein Adsorption Kinetics 847Kamal Al-Malah and Hasan Abdellatif Hasan MousaIndex 871vi ContentsContributorsZbigniew Adamczyk Institute of Catalysis and Surface Chemistry, Polish Academy ofSciences, Cracow, PolandKamal Al-Malah Department of Chemical Engineering, Jordan University of Science andTechnology, Irbid, JordanFerenc Berger Department of Colloid Chemistry, University of Szeged, Szeged, HungaryMalgorzata Boro wko Department for the Modelling of Physico-Chemical Processes, MariaCurie-Sklodowska University, Lublin, PolandGianfranco Cerofolini Discrete and Standard Group, STMicroelectronics, Catania, ItalyFrancisco Cuadros Departmento de Fisica, Universidad de Extremadura, Badajoz, SpainImre De ka ny Department of Colloid Chemistry, University of Szeged, Szeged, HungaryVladimir Nikolajevich Kislenko Department of General Chemistry, Lviv State PolytechnicUniversity, Lviv, UkraineSeung-Mok Lee Department of Environmental Engineering, Kwandong University,Yangyang, KoreaJohannes Lu tzenkirchen Institut fur Nukleare Entsorgung, Forschungszentrum Karlsruhe,Karlsruhe, GermanyHasan Abdellatif Hasan Mousa Department of Chemical Engineering, Jordan Universityof Science and Technology, Irbid, JordanAngel Mulero Departmento de Fisica, Universidad de Extremadura, Badajoz, SpainviiSalil U. Rege* Department of Chemical Engineering, University of Michigan, Ann Arbor,MichiganAlexander A. Shapiro Department of Chemical Engineering, Technical University ofDenmark, Lyngby, DenmarkErling H. Stenby Department of Chemical Engineering, Technical University of Denmark,Lyngby, DenmarkEtelka Tomba cz Department of Colloid Chemistry, University of Szeged, Szeged, HungaryJo zsef To th Research Institute of Applied Chemistry, University of Miskolc, Miskolc-Egyetemvaros, HungaryRalph T. Yang Department of Chemical Engineering, University of Michigan, Ann Arbor,MichiganLi Zhou Chemical Engineering Research Center, Tianjin University, Tianjin, China*Current afliation: Praxair, Inc., Tonawanda, New York.viii Contributors1Uniformand ThermodynamicallyConsistent Interpretation ofAdsorption IsothermsJOZSEF TOTH Research Institute of Applied Chemistry, University of Miskolc,Miskolc-Egyetemvaros, HungaryI. FUNDAMENTAL THERMODYNAMICS OF PHYSICAL ADSORPTIONA. The Main Goal of Thermodynamical TreatmentIt is well known that in the literature there are more than 100 isotherm equations derived basedon various physical, mathematical, and experimental considerations. These variances are justiedby the fact that the different types of adsorption, solid=gas (S=G), solid=liquid (S=L), andliquid=gas (L=G), have, apparently, various properties and, therefore, these different phenomenashould be discussed and explained with different physical pictures and mathematical treatments.For example, the gas=solid adsorption on heterogeneous surfaces have been discussed withdifferent surface topographies such are arbitrary, patchwise, and random ones. These models arevery useful and important for the calculation of the energy distribution functions (Gaussian,multi-Gaussian, quasi-Gaussian, exponential) and so we are able to characterize the solidadsorbents. Evidently, for these calculations, one must apply different isotherm equationsbased on various theoretical and mathematical treatments. However, as far as we know,nobody had taken into account that all of these different isotherm equations have a commonthermodynamical base which makes possible a common mathematical treatment of physicaladsorption. Thus, the main aim of the following parts of this chapter is to prove these commonfeatures of adsorption isotherms.B. Derivation of the Gibbs Equation for Adsorption on the Free Surface ofLiquids. Adsorption IsothermsLet us suppose that a solute in a solution has surface tension g (J=m2). The value of g changes asa consequence of adsorption of the solute on the surface. According to the Gibbs theory, thevolume, in which the adsorption takes place and geometrically is parallel to the surface, isconsidered as a separated phase in which the composition differs from that of the bulk phase.This separated phase is often called the Gibbs surface or Gibbs phase in the literature. Thethickness (t) of the Gibbs phase, in most cases, is an immeasurable value, therefore, it isadvantageous to apply such thermodynamical considerations in which the numerical value of t isnot required. In the Gibbs phase, ns1 are the moles of solute and ns2 are those of the solution, the1free surface is As (m2), the chemical potentials are ms1 and ms2(J=mol), and the surface tension is g(J=m2). In this case, the free enthalpy of the Gibbs phase, Gs(J), can be dened asGs= gAsms1ns1ms2ns2 (1)Let us differentiate Eq. (1) so that we havedGs= g dAsAs dg ms1 dns2ns1 dms1ms2 dns2ns2 dms2 (2)However, from the general denition of the enthalpy, it follows thatdGs= ssdT vsdP ms1 dns1ms2 dns2 (3)where ssis the entropy of the Gibbs phase (J=K) and vsis its volume (m3). Lete us compare Eqs.(2) and (3) so we get for constant values of As, T, and P;As dg ns1 dms1ns2 dms2 = 0 (4)The same relationship can be applied to the bulk phase with the evident difference that hereAs dg = 0that is,n1 dm1n2 dm2 = 0 (5)where the symbols without superscript s refer to the bulk phase.For the sake of elimination, let us multiply dm2 from Eq. (4) by ns2=n2 and take into accountthatdm1 =n2n1dm2 (6)and at thermodynamical equilibrium,dm1 = dms1 and dm2 = dms2 (7)so we have from Eq. (4),As dg ns1ns2n1n2 dms1 = 0 (8)The second term in the parentheses is the total surface excess amount of the solute material in theGibbs phase in comparison to the bulk phase. In particular, in the Gibbs phase, ns1 mol solute ispresent with ns2 mol solution, whereas in the bulk phase ns2(n1=n2) mol solute is present with n2mole solution. The difference between the two amounts is the total surface excess amount, ns1.So, according to the IUPAC symbols [1]ns1 = ns1ns2n1n2(9)Dividing Eq. (8) by As, we have the Gibbs equation, also expressed by IUPAC symbols, @g@m1 T= Gs1 (10)where the surface excess concentration (mol=m2) isGs1 =ns1As(11)2 To thIf the chemical potential of the solute material is expressed by its activity, that is,dm1 = RT d ln a1 (12)then the Gibbs equation (10) can be written in the practice-applicable formGs1 = a1RT@g@a1 T(13)In Eq. (13), the function g versus a1 is a measurable relationship because the activities of mostsolutes are known or calculable values, therefore, the differential functions, (@g=@a1)T, are alsocalculable relationships. So, we can introduce a measurable function cF(a1), dened ascF(a1) = a1@g@a1 T(14)Thus, the substitution of Eq. (14) into Eq. (13) yieldsGs1 = 1RTcF(a1) (15)The function cF(a1) has another and clear thermodynamical interpretation if it is written in theformcF(a1) = RTGs1 (16)Equation (16) is very similar to the three-dimensional gas law, namely, in this relationship,instead of the gas fugacity and the gas concentration (mol=m3), the function cF(a1)(J=m2) thesurface concentration (mol=m2), respectively, are present. It means that Eq. (16) can be regardedas a two-dimensional gas law.Equation (15) can also be considered as a general form of adsorption (excess) isothermsapplicable for liquid free surfaces. For example, let us suppose that the differential function ofthe measured relationship g versus a1 can be expressed in the following explicit form: dgda1= ab a1(17)where a and b are constants. So, taking Eq. (17) into account and substituting Eq. (14) into Eq.(15), we haveGs1 = 1RTaa1b a1(18)Equation (18) is the well-known Langmuir isotherm, applicable and measurable for liquid freesurfaces. It is evident that any measured and calculated explicit form of the function cF(a1)according to Eq. (15)yields the corresponding explicit excess isotherm equation.C. Derivation of the Gibbs Equation for Adsorption on Liquid=SolidInterfaces. Adsorption IsothermsThe derivation of the Gibbs equation for S=L interfaces is identical to that for free surfaces ofliquids if the following changes are taken into account:1. Instead of the measurable interface tension (g), the free energy of the surface, As(J=m2), is introduced and applied because, evidently, g cannot be measured on S=Linterfaces. From the thermodynamical point of view, there is no difference between Asand g.Interpretation of Adsorption Isotherms 32. In several cases, the surfaces As (m2) of solids cannot be exactly dened or measured.This statement is especially valid for microporous solids. According to the IUPACrecommendation [1], in this case the monolayer equivalent area (As;e) determined bythe BrunauerEmmettTeller (BET) method (see Section VI) must be applied. As;ewould result if the amount of adsorbate required to ll the micropores were spread in aclose-packed monolayer of molecules.Taking these two statements into account, instead of Eq. (8) the following relationship is validfor S=L adsorption when the liquid is a binary mixture:asm dAs ns1ns2n1n2 dms1 = 0 (19)where as is the specic surface area of the adsorbent (m2=g) (in most cases determined by theBET method), m (g) is the mass of that absorber and Asis the free energy of the surface. Here, itis also valid thatns1 = ns1ns2n1n2 (20)Dividing Eq. (19) by asm = As and applying again the relationship dm1 = RT d ln a1, we obtain a1RT@As@a1 T= ns1asm= Gs1 (21)If the function c(a1), similar to Eq. (14), is introduced, then we havecS;L(a1) = a1@As@a1 T(22)That is,Gs1 =cS;L(a1)RT(23)orcS;L(a1) = RTGs1 (24)From Eq. (21), it follows thatGs1 = ns1asm=n1sAs(25)Equation (25) denes the surface excess concentration, Gs1, where the surface of the solidadsorbent, in most cases, is determined by the BET method.In L=S adsorption, Eq. (23) or (24) cannot be applied directly for the calculation of theexcess adsorption isotherm because the function Asversus a1, as opposed to the function g versusa1, is not a measurable function. Therefore, another method is required to measure the excesssurface concentration; however, this measured value must be compared with the value of Gs1present in the Gibbs equation (24).The basic idea of this method is the following. Let the composition of a binary liquidmixture be dened by the mole fraction of component 1; that is,x1;0 = n1;0n1;0n2;0=n1;0n0; (26)4 To thwhere n1;0 and n2;0 are the moles of the two components before contacting with the solidadsorbent and n0 is the sum of the moles.When the adsorbent equilibrium is completed, the composition of the bulk phase can againbe dened by the mole fraction of component 1:x1 = n1n1n2= n1;0ns1n1;0ns1n2;0ns2= n1;0ns1n0 ns1ns2) (27)where ns1 and ns2 are the moles adsorbed into the Gibbs phase (i.e., these amounts disappearedfrom the bulk phase). From Eqs. (26) and (27), we obtainn0(x1;0x1) = ns1(1 x1) ns2x1 (28)The left-hand side of Eq. (28) includes measurable parameters only and is dened by therelationshipnn(s)1 = n0(x1;0x1) (29)where nn(s)1 is the so-called reduced excess amount, because nn(s)1 is the excess of the amount ofcomponent 1 in a reference system containing the same total amount, n0, of liquid and in which aconstant mole fraction, x1, is equal to that in the bulk liquid in the real system. Equations (28)and (29) were derived for rst time by Bartell and Ostwald and de Izaguirre [2, 3]. Theimportance of Eq. (29) is in the fact that it permits the measurement of the nn(s)1 versus x1 excessisotherms directly. However, the exact thermodynamical interpretation of S=L adsorptionrequires that the measured value of nn(s)1 in Eq. (29) be compared with the surface excessconcentration, Gs1, present in Gibbs equation (24). In order to this comparison, let us introduce inEqs. (28) and (29) the reduced surface excess concentration, (i.e., let us divide those relation-ships by As). Thus, we obtainGn(s)1 = A1s {ns1(1 x1) ns2x1} (30)whereGn(s)1 = A1s n1(s) = A1s {n0(x1;0x1)} (31)It has been proven by Eqs. (25) and (20) thatGs1 =ns1As= A1sns1ns2n1n2 (32)Let us write Eq. (32) in the form:Gs1 = A1sns1ns2x11 x1 (33)From Eqs. (30) and (33), we obtain the relationship between the reduced surface excesscontraction, Gn(s)1 , and the one present in the Gibbs equation (21):Gs1 = Gn(s)11 x1(34)Taking Eqs. (21) and (34) into account, we obtain the following Gibbs relationship:DAs1 = RTAs

a1(max)0Gn(s)1(1 x1)da1a1(35)Interpretation of Adsorption Isotherms 5Equation (35) provides the possibility for calculating the change in free energy of the surface,DAs1, if the activities of component 1 are known. In dilute solutions, a1 - x1; therefore, in thiscase, the calculation of DAs1 by Eq. (35) is very simple.The most complicated problem is to calculate or determine the composite (absolute)isotherms ns1 versus x1 and ns2 versus x2 because, in most cases, we do not have any informationabout the thickness of the Gibbs phase. If it is supposed that this phase is limited to a monolayer,then it is possible to calculate the composite isotherms.We can set out from the relationshipns1f1ns2f2 = As (36)where f1 and f2 are the areas effectively occupied by 1 mol of components 1 and 2 in themonolayer Gibbs phase (m2=mol). From Eqs. (36), (28), and (29), we obtain the compositeisothermsns1 = Asx1f2nn(s)1f1x1f2(1 x1) (37)andns2 =As(1 x1) f1nn(s)1f1x1f2(1 x1) (38)Equations (37) and (38) can be applied whenin addition to the monolayer thicknessthefollowing conditions are also fullled: (1) The differences between f1 and f2 are not greaterthan 30%, (2) the solution does not contain electrolytes, and (3) lateral and vertical interactiondo not take place between the components. In Fig. 1 can be seen the ve types of isotherm, nn(s)1versus x1, classied for the rst time by Schay and Nagy [4]. In Fig. 2 are shown thecorresponding composite isotherms calculated by Eqs. (37) and (38).It should be emphasized that the fundamental thermodynamics of S=L adsorption isexactly dened by (35) and are also the exact measurements of the reduced excess isothermsbased on Eq. (29). However, the thickness of the Gibbs phase (the number of adsorbed layers),the changes in the adsorbent structure during the adsorption processes, and interactions ofcomposite molecules in the bulk and Gibbs phases are problems open for further investigation.More of them are successfully discussed in Chapter 10.D. Derivation of the Gibbs Equation for Adsorption on Gas=SolidInterfacesThis derivation essentially differs from that applied for the free and S=L interfaces, because, inmost cases, the bulk phase is a pure gas (or vapor) (i.e., we have a one-component bulk andGibbs phase; therefore, the excess adsorbed amount cannot be dened as it has been taken in thetwo-component systems). This is why we are forced to apply the fundamental thermodynamicalrelationships in more detail than we have applied it earlier at the free and S=L interfaces.The rst law of thermodynamics applied to a normal three-dimensional one-componentsystem is the following:dU = T dS P dV m dn (39)where U is the internal energy (J), S is the entropy (J=K), V is the volume (m3), m is the chemicalpotential (J=mol), P is the pressure (J=m3), and n is the amount of the component (mol).6 To thFIG. 1 The ve types of excess isotherm nn(s)1 versus x1 classied by Schay and Nagy [4].Interpretation of Adsorption Isotherms 7FIG. 2 The composite monolayer isotherms corresponding to the ve types of excess isotherm andcalculated by Eqs. (37) and (38).8 To thLet us apply Eq. (39) to the Gibbs phase; thus, it is required to complete Eq. (39) with thework (J) needed ot make an interface; that is,dUs= T dSsP dVsmsdnsAsdAs (40)where the superscript s refers to the Gibbs (sorbed) phase (i.e., Usis the inside energy of theinterface, Ssis the entropy, and Asis the free energy of the interface [Gibbs phase]). Let usexpress the total differential of Eq. (40):dUs= T dSsSsdT P dVsVsdP msdnsnsdmsAsdAsAs dAs(41)Equations (40) and (41) must be equal, so we obtainnsdms= SsdT VsdP As dAs(42)Dividing both sides of Eq. (42) by ns, we have the chemical potential of the Gibbs phase:dms= ssdT vsdP Asns dAs(43)where ssand vsare the molar entropy and volume, respectively, of the Gibbs phase. Thechemical potential of the bulk phase (one-component three-dimensional phase) is equal to Eq.(43), excepted for the work required to make an interface. Thus, we obtaindmg= sgdT vgdP (44)where the superscript g refers to the bulk (gas) phase. The condition of the thermodynamicalequilibrium isdmg= dms(45)Taking Eqs. (43)(45) into account, we haveAs@As@P T= ns(vgvs) (46)Equation (46) is the Gibbs equation valid for S=G interfaces. As it can be seen, the thickness[i.e., the molar volume of the Gibbs phase (vs)] is an important parameter function here.On the right-hand side of Eq. (46), vgnsis the volume (m3) of nsin the bulk (gas) phaseand nsvsis the volume of nsin the Gibbs phase. It means that the differencens(vgvs) = Vs(47)is the surface excess volume of adsorptive (expressed in m3), which, according to the IUPACsymbols, is called Vs; that is, the exact form of Gibbs equation (46) isVs= As@As@P T(48)Let us express Eq. (48) as the surface excess amount (in mol), ns; it is necessary to divide Eq.(48) by the molar volume of the adsorptive, that is,ns=Vsvg (49)or, taking Eq. (47) into account,ns= ns

1 vsvg

(50)Interpretation of Adsorption Isotherms 9Thus, Eq. (48) can be written in the modied formns= Asvg @As@P T(51)Let us integrate Eq. (51) between the limits P and Pm, where Pm is the equilibrium pressure whenthe total monolayer capacity is completed. Thus, from Eq. (51) we obtainAs(P) = 1As

PmPnsvgdP (52)Suppose that the absorptive in the gas phase behaves like an ideal gas; we can then writeAs(P) =RTAs

PmPnsPdP (53)If the conditionvgvs(54)is fullled, then taking Eq. (50) into account, we obtainAs(P) =RTAs

PmPnsPdP (55)In spite of the simplications leading to Eq. (55), this relationship is the well-known and widelyused form of the Gibbs equation.It may occur that the absorptive in the gas phase does not behave as an ideal gas. In thiscase, instead of pressures, the fugacities should be applied or the appropriate state equationvg= f (P) (56)must be substituted in Eq. (55), that is,As(P) = 1As

PmPnsf (P) dP (57)Evidently, Eq. (57) is valid only if condition (54) is fullled. In the opposite case, the equationAs(P) = 1As

PmPnsf (P) dP (58)must be taken into account.E. The Differential Adsorptive PotentialThe Gibbs equations derived for free, S=L, and S=G interfaces provide a uniform picture ofphysical adsorption; however, they cannot give information on the structure of energy [i.e., wedo not know how many and what kind of physical parameters or quantities inuence the energy(heat) processes connected with the adsorption]. As it is well known these heat processes can beexactly measured in a thermostat of approximately innite capacity. This thermostat contains theadsorbate and the adsorptive, both in a state of equilibrium. We take only the isotherm processesinto account [i.e., those in which the heat released during the adsorption process is absorbed bythe thermostat at constant temperature (dT = 0) or, by converse processes (desorption), the heatis transferred from the thermostat to the adsorbate, also at constant temperature]. Under theseconditions, let dns-mol adsorptive be adsorbed by the adsorbent and, during this process, an10 To thamount of heat dQ (J) be absorbed by the thermostat at constant T. Thus, the general denitionof the differential heat of absorption is@Q@ns X;Y;Z= qdiff(59)where X; Y, and Z are physical parameters which must be kept constant for obtaining the exactlydened values of qdiff. Let us consider the parameters X = T, Y = vsand vg, and Z = As; we cannow discuss the problems of the adsorption mechanism as in Ref. 5.The molecules in the gas phase have two types of energy: potential and kinetic. During theadsorption process, these energies change and these changes appear in the differential heat ofadsorption. The potential energy of a molecule of adsorptive can be characterized by acomparison: A ball standing on a table has potential energy related to the state of a ball rollingon the Earths surface. This potential energy is determined by the character and nature of theadsorbent surface and by those of the molecule of the adsorptive.The kinetic energies of a molecule to be adsorbed are independent of its potential energyand can be dened as follows. Let us denote the rotational energy of 1 mol adsorptive as Ugrot andUsr is that in the adsorbed (Gibbs) phase. So, the change in the rotational energy isDUr = Ugr Usr (60)Similarly, the change in the translational energy isDUt = Ugt Ust (61)The internal vibrational energy of molecules is not inuenced by the adsorption; however, tomaintain the adsorbed molecules in a vibrational movement requires energy dened asDUsv = Usv Usv;0 (62)where Usv is the vibrational energy of 1 mol adsorbed molecules and Usv;0 is the vibrationalenergy of those at 0 K. If the above-mentioned potential energy is denoted by U0; then we obtainqdiffh = U0DUrDUtDUsv Usl (63)where the subscript h refers to homogeneous surface and Usl is the energy which can beattributed to the lateral interactions between molecules adsorbed. Equation (63) can be written ina shortened form if the two changes in kinetic energies are added:DUk = DUrDUt (64)that is, Eq. (63) can be writtenqdiffh = U0DUk DUsv Usl (65)The energy connected with the lateral interactions, Usl , depends on the coverage (i.e., the greaterthe coverage or equilibrium pressure, the larger is Usl . This is why the differential heat ofadsorption, in spite of the homogeneity of the surface, changes as a function of coverage (ofequilibrium pressure). However, in most cases, the adsorbents are heterogeneous ones; therefore,it is very important to apply Eq. (65) for these adsorbents too. For this reason, let us consider theheterogeneous surface as a sum of N homogeneous patches having different adsorptive potential,U0i (patchwise model). According to the known principles of probability theory, one can writeWi = Dditi(1 Yi) (66)where Wi is the probability of nding a molecule adsorbed on the ith patch, Ddi is the extent ofthe patch (expressed as a fraction of the whole surface), ti is the relative time of residence of theInterpretation of Adsorption Isotherms 11molecule on the ith patch, and Yi is the coverage of the same patch. In this sense, it can bedened an average or differential adsorptive potential, formulated as follows:Udiff0 =Ni WiU0;iNi Wi(67)Similar considerations yieldDUs;diffv =Ni WiDUsv;iNi Wi(68)Because the kinetic energies and Usl do not change from patch to patch (i.e., they areindependent of U0;i), we can writeqdiff= Udiff0 DUk DUsv; diff Usl (69)If the heterogeneity of the surface is not too small, then it can be estimated thatUdiff0 Uls DUk DUs;diffv (70)From relationship (70), it follows that the differential potential is approximately equal to thedifference between the differential heat of adsorption and the energy of lateral interactions; thatis,Udiff0 = qdiffUsl (71)As will be demonstrated in the next section, the thermodynamic parameter functions, AsandUdiff0 are the bases of a uniform interpretation of S=G adsorption. However, before thisinterpretation, a great and old problem of S=G adsorption should be discussed and solved.II. THERMODYNAMIC INCONSISTENCIES OF G=S ISOTHERM EQUATIONSA. The Basic Phenomenon of InconsistencyIn Section I.D., it has been proven that the exact Gibbs equation (48) contains the surface excessvolume, Vs, dened by the relationshipVs= ns(vgvs) (72)where nsis the measured adsorbed amount (mol) and vgand vsare the molar volume (m3=mol)of the measured adsorbed amount in the gas and in the adsorbed phase, respectively. Equation(72) means that nsshould be equal to the equationns=

Vsvgvs

(73)Let us calculate the function ns(P) for methane (i.e., for the methane isotherms by concretemodel calculations). From the literature [6], we obtain the following data. The critical pressure,Pc, is 4.631 MPa and the critical temperature, Tc, is 190.7 K. Thus, the reduced pressure (p) andreduced temperature (W) are p = P=Pc and W = 1:56 if the calculation is made for isotherms at298.15 K (25

C). Also from the literature [6], at W = 1:56 in the range of 8 _ p _ 30 (i.e.,37 MPa _ P _ 139 MPa), the compressibility factor Z varies approximately as a linear function:Z(p) = 0:0682p 0:356 (74)12 To thTaking into account thatvg= Z(p)RTP(75)we can calculate the molar volume of the gas phase in the pressure range 8 _ p _ 30. Thefunctions vg(P) can be seen in Fig. 3.Together with the function vg(P), is plotted the surface excess volume function Vs(P) iscalculated on the real supposition that in this range of pressure, Vs(P) decreases (see Fig. 4). Inthe left-hand side of Fig. 3, two linear functions Vs(P) are plotted:Vs(P) = 0:1 106P 45 (76)(Fig. 3, top) andVs(P) = 0:2 106P 45 (77)(Fig. 3, bottom), where P is expressed in MPa. Equations (76) and (77) mean that a smaller andgreater decreasing of Vs(P) have been taken into account. In the right-hand side of Fig. 3, thefunctions ns(P) can be seen. These functions have been calculated using Eq. (73), assumingdifferent values of vs(30 cm3=mol and 20 cm3=mol). Evidently, in the whole domain ofFIG. 3 Model calculations prove that in a high equilibrium pressure range, the gas=solid adsorptionisotherms have maximum values.Interpretation of Adsorption Isotherms 13pressure, vg> vsis valid. The functions ns(P) (i.e., the form of isotherms) demonstrate whereand why the measured adsorbed amount has the maximum value. The reality of this modelcalculation has also been proven experimentally by many authors published in the literature [7].The last of those is shown in Fig. 4 [8].As a summary of these considerations, it can be stated that according to the Gibbsthermodynamics, a plateau of isotherms in the range of high pressures, especially when P tendsto innity (P o), cannot exist.B. Inconsistent G=S Isotherm EquationsIn spite of the proven statements mentioned in Section II.A, there are many well-known andwidely used isotherm equations which contradict the Gibbs thermodynamics (i.e., theseequations are thermodynamically inconsistent). The oldest of these is the Langmuir (L) equation[9], having the following form:Y = P1=KLP(78)orP = 1KLY1 Y (79)whereY = nnsm(80)andKL = k1B exp U0RT (81)FIG. 4 Direct measurement proves that in a high pressure range, the adsorption isotherm of methanemeasured on GAC activated carbon at 298 K decreases approximately linearly. (From Ref. 8.)14 To thIn Eqs. (80) and (81), nsm is the total monolayer capacity, U0 is the constant adsorptive potential,and kB is dened by de Boer and Hobson [10]:kB = 2:346(MT)1=2105(82)where M is the molecular mass of the adsorbate and T is the temperature in Kelvin. Thenumerical values in Eq. (82) are correct if P is expressed in kilopacals.The inconsistent character of Eq. (78) or Eq. (79) appears in their limiting values. Inparticular,limPoY = 1 (83)orlimPYP = o (84)These limiting values mean that the total monolayer capacity is only completed if P tends toinnity (i.e., P decreases without limits while the isotherm has a plateau, as is shown in Fig. 5).In Section II.A, it has been proven that according to the Gibbs thermodynamics, a plateauin the range of great pressure cannot exist; therefore, the Langmuir equation is thermodynami-cally inconsistent. This statement is valid for all known and used isotherm equations havinglimiting values (83) or (84). The most important of those are discussed in Section III and it isdemonstrated there how this inconsistency can be eliminated in the framework of a uniforminterpretation of G=S adsorption.III. THE UNIFORM AND THERMODYNAMICALLY CONSISTENT TWO-STEPINTERPRETATION OF G=S ISOTHERM EQUATIONS APPLIED FORHOMOGENEOUS SURFACESThe elimination of the thermodynamical inconsistency of the isotherm equations can be done intwo steps. the rst step is a thermodynamical consideration and the second one is a mathematicaltreatment. Both can be made independently of one another; however, a connection exists betweenthem and this connection is the main base of the uniform and consistent interpretation of G=Sisotherm equations.FIG. 5 The Langmuir equation (78) is thermodynamically inconsistent because it has a plateau as thegreat equilibrium pressure goes to innity: nsm = 10:0 mmol=g, KL = 0:05 MPa1, Pm o.Interpretation of Adsorption Isotherms 15A. The First Step: The Limited Form and Application of the Gibbs EquationEquation (55) is the limited form of the Gibbs equation because it includes the suppositionsvgvsand the applicability of the ideal-gas law.Let us introduce in Eq. (55) the coverage dened by Eq. (80); we now obtainAs(P) = Asid

PmPYPdP (85)whereAsid =RTjm(86)In Eq. (86),jm = Asnsm(87)that is, jm is equal to the surface covered by 1 mol of adsorptive at Y = 1. It is easy to see thatEq. (86) is the free energy of the surface when the total monolayer is completed (ns= nsm) andthis monolayer behaves as an ideal two-dimensional gas. Therefore, Asid can be applied as areference value; that is,Asr(P) =As(P)Asid(88)So, from Eq. (85), we obtainAsr(P) =

PmPYPdP: (89)Equation (89) denes the change of the relative free energy of the surface, Asr(P), in thepressure domain P Pm. Equation (89) is thermodynamically correct if, in the pressure domainP Pm, the ideal-gas law is applicable and the supposition vgvsis valid. The applicability ofEq. (89) may be extended if instead of pressures, the fugacities are applied (i.e., the limits ofintegration are f and fm, corresponding to pressures P and Pm, respectively). This extension ofEq. (89) is supported by the fact that the supposition vgvsin most cases is still valid wheninstead of the ideal-gas state equation the relationship (56) should be applied.B. The Second Step: The Mathematical Treatment and the ConnectionBetween the First and Second StepsLet us introduce a differential expression having the formc(P) =nsP

dnsdP

1(90)It is important to emphasize that the numerical values of the function c(P) can be calculatedfrom the measured isotherm (viz. dns=dP is the differential function of the isotherm). It is alsoevident that this differential relationship can be calculated as a function of ns; that is,c(ns) =nsPdnsdP 1(91)16 To thThe values of functions (90) and (91) also do not change when the adsorbed amounts areexpressed in coverages, Y:c(Y) =YPdYdP 1(92)Let us write Eq. (92) in this form:dPP=c(Y)Y dY (93)From Eq. (93), we obtain

PmPdPP=

1Yc(Y)Y dY (94)orP = Pm exp

lYc(Y)Y dY (95)Similarly, integration of Eq. (90) yields

nsmnsdnsns =

PmPdPc(P)P(96)ornsnsm= Y = exp

PmPdPc(P)P (97)If the integration is performed between limits P P0 and Y Y0, where P0 is the saturationpressure and Y0 is the corresponding coverage, then we havePP0= Pr = exp

Y0Yc(Y)Y dY (98)andnsns0= Y0 = exp

P0PdPc(P)P (99)Equations (95) and (97)(99) are implicit integral isotherm equations with general validitybecause functions c(Y) or c(P) from any measured isotherms can be calculated. Theserelationships are only the results of a pure mathematical treatment. However, it is easy toprove the connection between the implicit integral isotherms and the limited Gibbs equation(89). In particular, let us substitute Eq. (92) into Eq. (89); then, we haveAsr(Y) =

Y0c(Y) dY (100)Equation (100) permits a simple numerical or analytical calculation of the relative change in freeenergy of the surface when the coverage changes in domains 0 Y. If Y is expressed in ns, thenAsr(ns) =RTAs

nsmnsc(ns) dns(101)Interpretation of Adsorption Isotherms 17From Eq. (100) follows the exact thermodynamical meaning of the function c(y):c(Y) = @Asr@Y T(102)It is important to emphasize again thatc(Y) = c(P) = c(ns) (103)when P, ns, or Y are conjugated pairs of the measured isotherms. It means that the functions inEq. (103) are thermodynamically equivalent. It is evident that the applicability of Eq. (100) andthe validity of Eq. (102) are equal to those of the Gibbs equation (89). Equations (95) and (97)(99) permit a consistent and uniform interpretation of G=S isotherms. The thermodynamicalconsistency is assured by the integration to a denite upper limit which can guarantee that theisotherm equations do not have limiting values equal to limits (83) or (86) and it is alsoguaranteed that all conditions leading to Eq. (55) are fullled. The uniformity assured that (1) allequations have the same implicit mathematical form, (2) in all equations, the functions c havingdirectly or indirectly the thermodynamical meaning dened by Eq. (102) and (3) the functions ccan be calculated from every measured isotherm so that can always be selected for theseisotherms the mathematically and thermodynamically correct equation. (For example see Fig. 18in Section III.H).C. The Uniform and Consistent Interpretation of the Modied LangmuirEquation, General ConsiderationsLet us apply the implicit integral relationships (96) for derivation of the Langmuir equation. Forthis reason, it must be demonstrated that the function c(P) belonging to the Langmuir equationiscL(P) =nsPdPdns = KLP 1 (104)Equation (104) can both be mathematically and experimentally proven; namely, if an isothermmeasured on a homogeneous surface is tted with a polinome and it is differentiated analytically,then the function cL(P) is equal to Eq. (104) (i.e., the slope of the straight line is equal to KL andthe zero point is equal to 1). Let us insert Eq. (104) into Eq. (97); we then obtainln nsnsm =

PPmdP(KLP 1)P(105)The integration on the right-hand side of Eq. (105) can be performed analytically:

PPmdP(KLP 1)P= ln KLPKLP 1 ln KLPmKLPm1 (106)Introducing the integration constant, wL,wL = 1 1KLPm(107)we obtain from Eqs. (106) and (107) the modied Langmuir (mL) equationns= nsmwLP1=KmLP(108)18 To thIn Eq. (109), the condition that at P = Pmnsis equal to the total monolayer capacity is fullled[i.e., the limiting values (83) and (84) are eliminated. The original Langmuir equation (78) doesnot contain the constant wL. Mathematically, this fact means that the integration in Eq. (97) isperfomed between the limits P and innity [i.e., the total monolayer capacity is completed at aninnitely great equilibrium pressure (wL = 1)].Before demonstrating other properties of the mL equation, it is necessary to prove thevalidity of Eq. (103); that is,c(ns) = c(P) = c(nY)It is easy to calculate the following function cm;L(Y) belonging to the modied Langmuirequation:cmL(Y) = wLwLY (109)Taking Eqs. (107) and (109) into account, we havewLwLY= KmLP 1 (110)It is important to remark that Eq. (104) belongs to the original Langmuir equation; however,from Eq. (110), we obtain the modied Langmuir equation (108).This result, demonstrated with the example of the mL equation, is of general validity andcan be drafted as it follows:1. If it is required to transform an inconsistent isotherm equation into a consistent one,then Eq. (97) or (99) should be applied, where c(P) belongs to the inconsistentequation.2. The function c(Y) of the inconsistent equation cannot be applied for this transforma-tion [i.e., the integration of Eq. (95) or (98) with the inconsistent functions c(y) doesnot lead to consistent isotherms equations]. However, the inconsistent functions c(y)are applicable to prove the inconsistency of the thermodynamical functions [see Eq.(112)].3. The reason for statement (1) is the fact that the function c(Y) has a concretethermodynamical meaning dened by Eq. (102). Therefore, all thermodynamicalconsistencies or inconsistencies are directly reected by the function c(Y).4. From statements (2) and (3), it follows that in Eqs. (95), (98), and (100), only theconsistent form of the function c(Y) can be applied.How these consistent forms of c(Y) can be calculated or determined are discussed in thefollowing subsections. However, before this discussion, it is required to demonstrate otherinconsistencies of the original Langmuir equation. The change in relative free energy of thesurface is dened by Eq. (100). To calculate this change, the explicit form of the function c(Y)is required. This function, belonging to the original Langmuir equation has the following form:cL(Y) = 11 Y (111)Let us substitute Eq. (111) into Eq. (100) and perform the integration; we thus obtainAsr(Y = 1) =

10dY1 Y= ln 11 Y 10= o (112)Interpretation of Adsorption Isotherms 19Equation (112) reects a thermodynamic inconsistency because the change in free energy of thesurface never can be innite. However, if we substitute the function cmL(y) [Eq. (109) into Eq.(100)], we haveDAsr(Y = 1)

10wLwLY dY = wL ln 1wLY 10= wL ln wLwL1 (113)Because wL > 1, the change in relative free energy of the surface always has a nite value. It isevident that if the integration in Eq. (100) is performed between the limits zero and a nite valueof Y, then we haveArs(Y) = wL ln wLwLY (114)Summarizing all considerations relating to the thermodynamic consistency of the mL equation, astatement of general validity can be made: The consistent form of the mL equation (and others)can be derived because Eq. (97) requires integration with a nite value of the upper limit. If thisupper limit, Pm, is not so great that instead of ns, the surface excess volume, Vs, or surfaceexcess amount, ns, ought to apply, then, according to Eq. (102), the thermodynamicalinterpretation of the function c(Y) is correct. Therefore, the isotherm equations derived fromEq. (97) or from Eq. (98) are also thermodynamically consistent. From this statement, it followsthat the inconsistencies of the well-known monolayer isotherm equations are such that theoriginal Langmuir equation and all those discussed in following sections are connected with thefact that these relationships were not derived from consistent differential equations requiringintegration. Thus, these relationships include the limiting valuelimPoY = 1which is thermodynamically inconsistent. It is also proven that all inconsistencies are reectedby the function c(Y). The discussions and relationships proving the consistency of the mLequation, inconsistencies of the original Langmuir equation, and Eqs. (95)(100) providing thederivation of consistent isotherm relationship permit a general method for interpretation of anyisotherm equation.The calculation method of this interpretation is demonstrated in detail with the example ofthe mL equation; however, this method can be (and is) applied for every isotherm equationdiscussed in this chapter. The results of these calculations are shown in Fig. 6 and details of thoseare in particular the following. For Fig. 6 (top, left), the applied mL equations areP = 1KmLYwLY (115)where wL and KmL may be varied; wL = 1 (solid line) is the original Langmuir isotherm. Thisgure represents the measured adsorption isotherms, assuming that the total monolayer capacity(nsm) is known (see BET method and others). For Fig. 6 (top right), the functions cmL(P) arelinear,cmL(P) = KmLP 1 (116)and it can be calculated directly because the mL equation can also be expressed in the formY = f (P). However, most of the isotherm equations cannot emplicitly be expressed in this form,20 To thas only the term P = W(Y) exists. In these cases, the calculation of the function c(P) is thefollowing. First, the function c(Y) is calculated, which, in this case, has the formcmL(Y) = wLwLY (117)FIG. 6 The uniform and thermodynamically consistent interpretation of the mL equation. The values ofthe parameters are as follows: KL = 0:1 kPa1, wL = 1:0, Pm o (solid line, original Langmuirequation); KmL = 0:04 kPa1, wL = 1:03, Pm = 833:3 kPa ( ); KmL = 0:15 kPa1, wL = 1:06,Pm = 111:11 kPa ( ); KmL = 0:08 kPa1, wL = 1:16, Pm = 78:13 kPa ( ).Interpretation of Adsorption Isotherms 21BecausecmL(Y) = cmL(P) (118)to every value of P calculated by Eq. (115) can be attributed a value of cmL(Y). Thus, we obtainthe function cmL(P) numerically. It is evident that if the isotherm investigated can be expressedboth in terms of Y = f (P) and P = j(Y), then the two methods lead to the same relationship,c(P).The practical importance of the function c(P) is discussed in Section III.A. For Fig. 6(middle, left), this interpretation of the measured isotherms is thermodynamically exact, becauseif reects the fact that the integration in Eq. (97) has been performed to the nite upper limit Pmand, therefore, the equilibrium pressure should be expressed in a relative pressure dened asPr;m = PPm(119)The value of Pm can be calculated form the integration constant wL dened by Eq. (107). Thus,we obtainPm = |KL(wL1)]1(120)Here, it can also be seen that at wL = 1, Pm o. This limiting value is thermodynamicallyinconsistent.For Fig. 6 (middle, right), the function c(Y) is very important form two standpoints. First,the analytical or numerical integration of this function permits the calculation of the relative freeenergy of the surface [see Eq. (100)]:Asr(Y) =

Y0c(Y) dY (121)Second, the function c(Y) is required to calculate the function c(P) numerically if the isothermequation cannot be expressed in terms of Y = f (P). In Fig. 6 (bottom, left), the functions Asr(Y)calculated analytically or numerically by Eq. (121) are represented. In Fig. 6 (bottom, right) thefunctions Asr(Pr;m) calculated similar to the function c(P) are shown. In particular, to every valueof Pr;m = P=Pm calculated by Eqs. (115) and (120) are attributed the values of Asr(Y), so weobtain the functions Asr(Pr;m). These two types of function in the bottom of Fig. 6 characterizethermodynamically the adsorption process and thus seem to complete the uniform interpretationof the mL and other isotherm equations. The thermodynamic consistency is best reected by thefunctions Asr(Y) and Asr(Pr;m) because both functions have nite values at Y = 1 or at Pr;m = 1.D. The Uniform and Consistent Interpretation of the ModiedFowlerGuggenheim EquationFowler and Guggenheim [11] derived an isotherm equation which takes the lateral interaction ofthe adsorbed molecules into account. It has the following explicit form:P = 1KFY1 Y exp(BFY) (122)whereBF =CoRT(123)22 To thIn Eq. (123), o is dened as the interaction energy per pair of molecules of nearest neighbors,and C, is a constant. Thus, the orignal Langmuir equations is transformed into Eq. (122).Equation (122) contains all thermodynamic inconsistencies mentioned in connection with theoriginal Langmuir equation because the limiting valueslimPoY = 1 and limy1P = oare also valid for Eq. (122).To obtain a consistent form of Eq. (122), let us calculate its function c(y):cF(Y) = 11 YBFY (124)It has been proven that the consistent Langmuir equation has the function cmL(Y) dened by Eq.(109); therefore, the consistent (modied) FowlerGuggenheim (mFG) equation should have thefollowing function:cmF(Y) = wFwF YBFY (125)Let us substitute Eq. (125) into Eq. (95). After integration, we haveP = Pm(wF 1) exp(BF) YwF Y exp(BFY) (126)where the constant of integrations, IF, isIF = Pm(wF 1) exp(BF) (127)Let us compare Eqs. (126) and (127) with Eq. (122); we haveP = 1KmFYwFY exp(BFY) (128)whereKmF = I1F = |Pm(wF 1) exp(BF)]1(129)orPm = IF|(wF 1) exp(BF)]1(130)Therefore, the consistent form of the FG equation is relationship (128). For example, the limitingvalue of Eq. (128) islimY1P = Pm (131)It is expected that the original and modied FG equations, which take explicitly the lateralinteractions between molecules adsorbed into account, can describe different types of isotherms.In Fig. 7 can be seen the four types of isotherm which the modied FG equation can describeand explain. In the following, the limits and values of parameters (wF; BF) determining the typesof isotherms are interpreted.From the physical meaning of the mFG relationships, it follows that they have to reect thetwo-dimensional condensation too, similar to the three-dimensional van der Waals equation. Inthis case, the isotherm equation,P = j(Y)Interpretation of Adsorption Isotherms 23should have local maximum and local minimum values. These values of y exist when theconditiondj(Y)dY = 0 (132)is met. Let us differentiate Eq. (128) and calculate the values of BF for which Eq. (132) isfullled; we thus obtainBF = wFY(wF Y) (133)Equation (133) determines all values of BF at which two-dimensional condensation takes place.The coverages, Y, present in Eq. (133) are the places of minima and maxima mentioned earlier(see also the S-shape condensation isotherm in Fig. 7). The functions BF(Y) are shown in Fig. 8by solid lines.In Fig. 8, it can be seen also that functions BF(Y) have absolute minimum values. Afterdifferentiation of Eq. (133), we have the values of coverage where these minima occur:Ymin = 0:5wF (134)How these places of minima, Ymin, increase according to Eq. (134) are shown in Fig. 8; however;by inserting Eq. (134) into Eq. (133), the decreasing character of BF;min can be calculatedexplicitly:BF;min = 4wF(135)In Fig. 8, the values of BF;min are represented by horizontal dotted lines.For the determination of other types of isotherm corresponding to Eq. (128), it is essentialto calculate the function BF(Y) which fulls the conditioncmF(Y) = 1 (136)According to Eq. (125), condition (136) is met by the following values of BF and Y:BF = 1wF Y (137)In Fig. 8, functions (137) are shown with dash-dot-dot lines.FIG. 7 The modied FowlerGuggenheim equation can describe four types of isotherm (Types I, III, andV, and condensation).24 To thMathematically, the condition c(Y) = 1 means that all values of Y present in Eq. (137)can be drawn from the origin proportional lines. One of these situations is represented in Fig. 9.The regions of coverages where c(Y) > 1 and c(Y) < 1 and the point where the proportionalline drawn from the origin is a tangent can be seen. Evidently, c(Y) = 1 is also valid when theinitial domain of an isotherm is a proportional line (i.e., the isotherm begins with a Henrysection).The above analysis is also represented in Fig. 8. The rst gure (wF = 1) relates to theoriginal FG equation, which can describe Types I and V and condensation isotherms. However,for Type V isotherms, the value of BF tends to innity when Y tends to 1. In this fact is alsoreected the thermodynamical inconsistency of the original FG equation. In the top (right) ofFig. 8, wF = 1:17, the place of minimum, Ymin, according to Eq. (134) has been increased. So,from the analysis and gures above, it follows that the Type I isotherm is described when0 < BF < 1=wF, Type V isotherms can occur when 1=wF < BF < 4=wF, and two-dimensionalcondensation takes place when BF > 4=wF. A very interesting limiting case is shown at theFIG. 8 Limiting values of parameters BF and wF determining the types of FG and mFG isotherm.Interpretation of Adsorption Isotherms 25bottom (left) of Fig. 8 when wF = 1:333: In this case, the minimum value of BF;min; that is, Eq.(134) at Y = 1 is equal to the value of BF corresponding to Eq. (137):4wF= 1wF 1 (138)Solving Eq. (138) for wF, we obtainwF = 1:333 (139)Equation (139) means that if the values of wF are greater than 1.333, then the modied FGequation can also describe isotherms of Type III. This situation is represented at the bottom(right) of Fig. 8 when wF = 2:00. Thus, the extended applicability of the modied FG equation isthe following. If wF is greater than 1.333, then the types of isotherm are determined by thefollowing limiting values of BF:Type I when 0 < BF < 1=wFType V when 1=wF < BF < 1=(wF 1)Type III when 1=(wF 1) < BF < 4=wFType condensation when BF > 4=wFThus, it is proven why the four types of isotherm shown in Fig. 7 can be described by themodied FG equation.The analysis made above is limited to mathematical considerations. In Figs. 1012 thethermodynamically consistent and uniform interpretation of isotherms Types I, III, and V,respectively, are shown. In these gures, quite equal to Fig. 6, are represented the functionsY(P), c(P), Y(Pr;m), c(Y), Asr(Y), and Asr(Pr;m). The calculations of these functions have beenmade as follows:Top (left): The applied mFG equations areP = 1KmFYwF Y exp(BFY) (140)The types of isotherms in Figs. 1012 have been determined by the corresponding valuesof wF and BF (see Fig. 8).FIG. 9 The values of function c(Y) corresponding to the Type V isotherm.26 To thFIG. 10 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type I.The values of the parameters are as follows: KF = 0:08 kPa1, wF = 1:0, BF = 0:8, Pm o (solid line,original FG equation); KmF = 0:04 kPa1, wF = 1:03, BF = 0:7, Pm = 413:8 kPa ( );KmF = 0:06 kPa1, wF = 1:04, BF = 0:5, Pm = 252:7 kPa ( ); KmF = 0:08 kPa1, wF = 1:05,BF = 0:8, Pm = 112:3 kPa ( ).Interpretation of Adsorption Isotherms 27FIG. 11 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type III.The values of the parameters are as follows: KmF = 15 104kPa, wF = 2:2, BF = 1:1, Pm = 184:9 kPa( ); KmF = 8 104kPa, wF = 2:8, BF = 1:50, Pm = 155:0 kPa ( ); KmF = 11 104kPa,wF = 3:2, BF = 1:25, Pm = 118:4 kPa ( ).28 To thFIG. 12 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type V.The values of parameters are as follows: KF = 6 103kPa1, wF = 1:0, BF = 3:0, Pm o(solid line,original FG equation); KmF = 0:01 kPa1, wF = 1:03, BF = 2:5, Pm = 273:6 kPa ( );KmF = 8 103kPa1, wF = 1:05, BF = 2:0, Pm = 338:3 kPa ( ); KmF = 3 103kPa1,wF = 1:07, BF = 3:0, Pm = 237:1 kPa ( ).Interpretation of Adsorption Isotherms 29Top (right): The functions c(P) by the relationshipscmF(Y) = wFwF YBFY (141)andcmF(Y) = cmF(P) (142)Middle (left): The thermodynamically consistent and uniform interpretation of themeasured isotherms in form of Y(Pr;m); that isPr;m = PPmandPm = K1mF|(wF 1) exp(BF)]1(143)have been calculated.Middle right: The functions c(Y) are bases for calculation of functions Asr(Y) and Asr(Pr;m):Bottom: The functions Asr(Y) and Asr(Pr;m) are calculated numerically by the integralequationAsr(Y) =

Y0c(Y) dY (144)These functions (i.e., the relative changes in free energy of the surface shown in Figs. 1012)characterize thermodynamically the adsorption processes. In particular, in Fig. 11, (isotherms ofType III)Asr(Y) < 1 (145)is valid in the whole domain of coverage; that is,Asr(Y) < Asid (146)This means that the change in free energy of the surface is always less than that would have beencaused by a two-dimensional ideal monolayer completed on a homogeneous surface. However,for isotherms of Type I and V (in Figs. 10 and 12, respectively), there always exists a denitecoverage whereAsr(Y) = Asid (147)The values of Y and Pr;m when Eq. (147) is valid are excellent and very simple parameterscharacterizing the adsorption system investigated.For practical applications of the mFG equation and for calculations of constants Km;F, wF,BF, and Pm, a three-parameter tting procedure is recommended. In particular, Eq. (126) can bewritten in the formP = IFYwF Y exp(BFY) (148)Equation (148) can be tted to the measured points (Y; P) with parameters IF, wF, and BF. Theaverage percentile deviation (D%) has been calculated using the following relationship:D%= 1NNi=1(P PcP100

(149)30 To thwhere P is the measured equilibrium pressure, Pc is calculated equilibrium pressures, and N isthe number of the measured points (Y; P).We have the constants IF, wF, and BF as results of the tting procedure. According to Eq.(129), we obtainKmF = (IF)1(150)Finally, the pressure when the total monolayer capacity is completed, Pm yields Eq. (130) or(127),Pm = IF|(wF 1) exp(BF)]1(151)E. The Uniform and Consistent Interpretation of the Modied VolmerEquationVolmer [12] was the rst scientist to take the mobility of the adsorbed molecules into account.His considerations were based on the dynamic equilibrium between the gas and adsorbed phaseand obtained the following relationship:P = 1KVY1 Y exp Y1 Y (152)where the exponential term reects the mobility of molecules in the adsorbed layer. The functionc(Y) of the Volmer equation has the formcV(Y) = 11 Y 2(153)Equation (153) means that the mobility of the monolayer, in comparison to the immobileLangmuir monolayer, is expressed by the relationshipcV(Y) = |cL(Y)]2(154)Unfortunately, Eq. (152) is thermodynamically inconsistent, because in it, the limiting valueslimPoY = 1 and limY1P = oare on contradiction with the Gibbs equation (51) and it is also unacceptable that according toEq. (100),limY1Asr = o (155)namely,Asr(Y = 1) = 1

1YdY(1 Y)2 = o (156)These are the reasons why a consistent form of the Volmer equation should be derived. Thisderivation is quite similar to Eqs. (109) and (125); that is, instead of Eq. (153), a modiedVolmer (mV) equation can be written:cmV(Y) = wVwV Y 2(157)Interpretation of Adsorption Isotherms 31Let us substitute Eq. (157) into Eq. (195); after integration, we obtainP = Pm(wV 1) exp 1wV 1 YwV Y exp YwV Y (158)where the constant of integrations, Iv, has the formIV = Pm(wV 1) exp 1wV 1 (159)Thus we obtain the modied Volmer equation in a simple form:P = 1KmVYwV Y exp YwV Y (160)whereKmV = (IV)1= Pm(wV 1) exp 1wV 1 1(161)orPm = IV (wV 1) exp 1wV 1 1(162)For practical applications of Eq. (158) and for calculations of constants KmV, wv, Pm, and nsm, athree-parameter tting procedure is proposed. In particular, Eq. (158) can be written asP = IVYwV Y exp YwV Y (163)orP = IVnswVnsmns exp nswVnsmns (164)The parameters to be tted are IV, wV, and nsm. The constants KV and Pm can be calculated similarto Eqs. (150) and (151),KmV = (IV)1(165)and equal to Eq. (162),Pm = IV (wV 1) exp 1wV 1 1(166)It is very important to remark that the original and the modied Volmer equation can onlydescribe isotherms of Type I. It is obvious, because, from Eq. (157), it follows thatwV > wV Y (167)that is, in the whole domain of coverage, it is valid thatcmV(Y) > 1 (168)This means that the mobility of adsorbed molecules cannot cause a change in the type ofisotherm. This change may only happen if the interactions between the adsorbed molecules aretaken into account, as it is done by the mFG equation.32 To thThe mathematically uniform and thermodynamically consistent interpretation of the mVequation is represented in Fig. 13. The functions Y(P), c(P), Y(Pr;m), c(Y), Asr(Y), and Asr(Pr:m)are calculated the same way as the calculations were performed for the mL and mFG equationsshown in Figs. 6 and 10. The most important difference between the interpretation, of mFG andmVequations is the functions c(P) (top right in Figs. 10 and 13). These functions, reect best thedifferences between interactions of the adsorbed molecules and the mobility of them.FIG. 13 The uniform and thermodynamically consistent interpretation of the mV isotherm of Type I. Theparameters are as follows: KV = 0:50 kPa1, wV = 1:0, Pm o (solid line, original Volmer equation);KmV = 0:50 kPa1, wV = 1:3, Pm = 186:9 kPa ( ); KmV = 0:30 kPa1, wV = 1:4, Pm = 101:5 kPa( ); KmV = 0:10 kPa1; wV = 1:5, Pm = 147:8 kPa ( ).Interpretation of Adsorption Isotherms 33F. The Uniform and Consistent Interpretation of the Modiedde BoerHobson EquationBoth the mobility and the interactions are taken into account by the de BoerHobson (BH)equation [10], having the following explicit form:P = 1KBY exp Y1 YBBY (169)The corresponding function cB(Y) iscB(Y) = 11 Y 2BBY (170)Equation (170) is also thermodynamical inconsistent because the limiting value is Y = 1 ifP o, with a plateau on the isotherms. To derive the consistent form, the modied form of thefunction cB(Y) is also required:cmB(Y) = wBwBY 2BBY (171)Substitution of Eq. (171) into Eq. (95) and integration yieldsP = Pm(wB1) exp BB 1wB1 YwBY exp YwBYBBY (172)where the constant of integration has the formIB = Pm(wB1) exp BB 1wB1 (173)Thus, we obtain the modied BH (mBH) equation in a simple form:P = 1KmBYwBYexp YwBYBBY (174)whereKmB = (IB)1= Pm(wB1) exp BB 1wB1 1(175)orPm = IB(wB1) exp BB 1wB1 1(176)Because Eq. (174) takes the interactions between the adsorbed molecules into account, it is againexpected that the mBH equation can describe different types of isotherms and reect the two-dimensional condensation also. The limits and values of BB and wB determining the applicabilityof Eq. (174) to different types of isotherm can be calculated quite similar to the limits and valuesshown in Fig. 8. However, the numerical values of relationships corresponding to the mBHequation differ from those plotted in Fig. 8.The starting point of our considerations is again the calculation of the values of BB whichmeet the conditiondj(Y)dY = 034 To thThus, we obtainBB = wBwBY 2Y1(177)Equation (177) corresponds to Eq. (133).The absolute minimum points, corresponding to Eq. (134) areYmin = 0:3333wB (178)The minima of BB, corresponding to Eq. (135) areBB;min =6:75wB(179)The values of BB which meet the condition cB(Y) = 1 areBB = 2wBY(wBY)2 (180)Equation (180) corresponds to Eq. (137).According to Eqs. (177)(180), the limits and values of BF mentioned are the following.The mBH equation describes isotherms of Type I when 0 < BB < 2wB. Isotherms of Type V canoccur when 2=wB < B < 6:75=wB, and two-dimensional condensation takes place whenBB _ 6:75=wB.Similar to Eq. (138), it may occur that6:75wB= 2wB1(wB1)2 (181)Solving the second-power equation (181), we obtainwB = 1:873 (182)This means that for all values of wB which are greater than 1.873, isotherms of Type III also aredescribed by the mBH equation. So, the limits and values of BB and wB are modied ifwB > 1:873: isotherms of Type I when 0 < BB < 2=wB; isotherms of Type V when2=wB < BB < (2wB1)=(wB1)2; and those of Type III when (2wB1)=(wB1)2< BB < 6:75=wB. These limits are interpreted in Fig. 14.For practical applications of Eq. (172) and for calculations of constants KmB, wB, BB, andPm, a three-parameter tting procedure is again recommended. In particular, Eq (172) has thefollowing form:P = IBYwBY exp YwBYBBY (183)Equation (183) can be tted to the measured points (Y; P), with parameters IB, wB, and BB.Therefore, as for Eqs. (150), (151), (165), and (166), we obtainKmB = (IB)1(184)andPm = IB (wB1) exp BB 1wB1 1(185)If the value of nsm is not known (i.e., Y cannot be calculated), then a four-parameter ttingprocedure may be tried. However, the calculation of nsm is discussed in detail in Section VI.Interpretation of Adsorption Isotherms 35The mathematically uniform and thermodynamically consistent interpretation of the mBHequation are shown in Figs. 1517. The function Y(P), Y(Pr;m), c(Y), Asr(Y), and Asr(Pr;m) inthese gures are similar to the corresponding relationships in Figs. 1012. The essential andimmediately observable difference can be seen in the function c(P) (see Figs. 10 and 15, topright). The importance of this difference is discussed in Section III.H.G. Physical Interpretation of Constants Kx Present in the ModiedIsotherm Equations Applied to Homogeneous SurfacesEquations (120), (129), (161), and (175) mathematically dene the constants KL, KmF, KmV, andKmB present in the corresponding isotherm equations. In addition to this interpretation, aphysical one is possible. According to de BoerHobsons theory [10], the Henry constant (H) ofan isotherm measured on homogeneous surface isH = k1B exp U0RT (186)where kB is dened by Eq. (83).FIG. 14 Limiting values of the parameters BB and wB determining the types of BH and mBH isotherms.36 To thThe original Langmuir equation has the formP = 1KLY1 Y (187)The Henry constant (H) is dened by the limiting valuelimP0YP= H (188)FIG. 15 The uniform and thermodynamically consistent interpretation of the mBH isotherms of Type I.The values of parameters are as follows: KB = 0:50 kPa1, wB = 1:00, BB = 1:50, Pm o (solid line,original BH equation); KmB = 1:1 kPa1, wB = 1:20, BB = 1:40, Pm = 166:4 kPa ( );KmB = 0:1 kPa1, wB = 1:30, BB = 1:40, Pm = 230:4 kPa ( ); KmB = 0:15 kPa1, wB = 1:4,BB = 0:40, Pm = 136 kPa ( ).Interpretation of Adsorption Isotherms 37It means that in the original Langmuir equation,KL = H (189)(i.e., KL represents the Henry constant). At the modied Langmuir equation,P = 1KmLYwLY (190)FIG. 16 The uniform and thermodynamically consistent interpretation of the mBH isotherms of Type III.The values of parameters are as follows: KmB = 0:75 103kPa1, wB = 3:1, BB = 1:70, Pm = 186:7 kPa( ); KmB = 103kPa1, wB = 3:0, BB = 1:60, Pm = 166:4 kPa ( ); KmB = 1:1 103kPa1,wB = 3:2, BB = 1:50, Pm = 145:3 kPa ( ).38 To thFIG. 17 The uniform and thermodynamically consistent interpretation of the mBH isotherms of Type V.The values of parameters are as follows: KB = 0:012 kPa1, wB = 4:50, Pm o(solid line, original BHequation); KmB = 0:012 kPa1, wB = 1:2; BB = 4:50, Pm = 687:0 kPa ( ); KmB = 2 103kPa1,wB = 1:4, BB = 4:20, Pm = 228:4 kPa ( ); KmB = 2:1 103kPa1, wB = 1:3, BB = 4:00,Pm = 815:0 kPa ( ).Interpretation of Adsorption Isotherms 39the limiting value isH = limP0YP= KmLwL = k1B exp U0RT (191)that is,KmL = (kBwL)1exp U0RT (192)Taking Eqs. (140), (160), and (174) into account, we also obtain the physical interpretation ofconstants, KmF, KmV, and KmB, respectively:KmF = (kBwF)1exp U0RT (193)KmV = (kBwV)1exp U0RT (194)KmB = (kBwB)1exp U0RT (195)It is easy to see and explain that these constants, representing the slopes of isotherms in very lowequilibrium pressures, differ in value from the parameter w only if the absorptive potential, U0,and kB are identical.H. Properties of the Function c(P) Corresponding to the ModiedLangmuir, FG, Volmer, and BH Isotherm EquationsIn previous subsections, the thermodynamic properties, especially the change in relative freeenergy of the surface, have been discussed. However, the calculation of the functions Asr(Y) andAsr(Pr;m) require integration according to Eq. (100); therefore, it needs time. This time isnecessary when we are interested in the thermodynamic properties of the adsorbed phase.Nevertheless, after measuring an isotherm, the following question should be answered imme-diately: Which isotherm equation can be applied to describe and explain the measured data? Thisproblem may be solved by calculation the function c(P) dened by relationship (90):c(P) =nsPdnsdP 1This calculation can be made without knowledge of the specic area of the absorbent and itneeds only the differentiation of an explicit function tted to the measured points. These functionc(P) corresponding to the modied Langmuir, FG, Volmer, and BH equations are shown inFig. 18.In the top (left) of Fig. 18 are plotted the functions c(P) corresponding to isotherms TypeI, which can be seen in the top (right) of Fig. 18. These isotherms of Type I are very similar;therefore, with a simple tting procedure the thermodynamically correct isotherm equationcannot be selected. However, the function c(P), especially in lower domain of pressure (seebottom in Fig. 18), are very different and characteristic for the corresponding isotherm equation.So, after calculation of the function c(P), the correct isotherm equation can be selected andapplied for the isotherm measured on a homogeneous surface.40 To thIV. THE UNIFORM AND THERMODYNAMICALLY CONSISTENTINTERPRETATION OF G=S ISOTHERM EQUATIONS APPLIED FORHETEROGENEOUS SURFACESA. The To th EquationAn old problemof adsorption theories is howthe very complexeffects of energetic heterogeneity ofsolids can be taken into account. One of these attempts have been made by Toth [13]. The startingpoint of his theory is the following observation. Aheterogeneous surface uptakes more adsorptive,and at the same, relative equilibrium pressure, than a homogeneous surface with specic surfacearea equal to that of the heterogeneous adsorbent. Consequently, in this case, there is no differencein the monolayer capacities (nsm) of the homogeneous and heterogeneous surfaces. This require-ment can be taken by one parameter, t, applied to the coverage as a power into account; that is,Yt> Y if 0 < t < 1 (196)It is also an experimental observation that if the lateral interactions between the adsorbedmolecules are greater than the adsorptive potential at the same coverage, thenYt< Y for t > 1 (197)FIG. 18 Comparison of isotherms of Type I and the corresponding functions c(P) calculated by the mL,mV, mFG and mBH equations. The values of parameters are as follows: mL: KmL = 0:04 kPa1, wL = 1:03,Pm = 833:3 kPa; mV: KmV = 0:50 kPa1, wV = 1:30, Pm = 187:0 kPa; mFG: KmF = 0:08 kPa1,wF = 1:05, Pm = 112:3 kPa; mBH: KmB = 0:15 kPa1, wB = 1:40, Pm = 136:1 kPa.Interpretation of Adsorption Isotherms 41First, let us consider relationship (196). According to the modied Langmuir equation, itsfunction c(Y) can be dened by Eq. (129):cmL(Y) = wLwLYTaking requirement (196) into account, we obtaincT(Y) = wTwT Yt 0 < t < 1 (198)Substitution of Eq. (198) into Eq. (95) and integration yieldsP = Pm(wT 1)1=t Y(wT Yt)1=t (199)where the constant of integrations isIT = Pm(wT 1)1=t(200)The simple form of Eq. (199) isP = 1KT 1=tY(wT Yt)1=t (201)whereKT = (IT)t(202)andPm = IT|(wT 1)1=t]1(203)It is evident that at Y = 1 and P = Pm.Equation (199) is thermodynamically consistent, because the change in relative free energyof the surface, when the total monolayer capacity is completed, is a nite value; namely,according to Eq. (100),Asr(Y = 1) =

10wTwT Yt dY = finite value (204)The constant KT can also be expressed by physical parameters. In particular, it is known that forhomogeneous surfaces, Eqs. (186) and (188) are valid; that isH = limP0YP= k1B exp U0RT (205)where U0 is the constant adsorptive potential. However, Eq. (199) relates to heterogeneoussurfaces; therefore, instead of U0, the corresponding value of the differential adsorptive potential,Udiff0 , should be present in Eq. (205). In the domain of the very low equilibrium pressures andcoverages (i.e., when P and Y tend to zero), it is also valid thatlimY0Udiff0 (Y) = const (206)In this sense, for heterogeneous surfaces, it is also correct thatH = limP0YP= k1B exp|Udiff0 (Y = 0)] (207)42 To thFor Eq. (201), it is valid thatH = limP0YP= (KTwT)1=t(208)So, comparing Eq. (208) with Eq. (207), we obtain(KTwT)1=t= k1B exp Udiff0 (Y = 0)RT (209)that is,KT =ktBwTexp tUdiff0 (Y = 0)RT (210)For practical applications of Eq. (201), a three-parameter tting procedure is proposed. Theparameters to be tted are KT, wT, and t. The value of Pm is calculated from Eq. (203). For thecalculation of the total monolayer capacity, a four-parameter iteration may also be possiblebecause Eq. (201) may also be written in this form:P = 1(KT)1=tns|wT(nsm)t(ns)t]1=t (211)However, for calculating nsm; a general method is discussed in Section VI.The mathematically uniform and thermodynamically consistent interpretation of Eq. (199)is shown in Fig. 19, where the functions Y(P), c(P), Y(Pr;m), c(Y), Asr(Y), and Asr(Pr;m) areplotted. The calculations of these functions have been made as follows:Top (left):P = 1KT 1=tY(wT Yt)1=t (212)orY = (wT)1=tP|(1=KT) Pt]1=t (213)Top (right):cT(P) = KTPt1 (214)or the values of the functioncT(Y) = wTwT Yt (215)are conjugated with values of P caculated from Eq. (212).Middle (left): From Eq. (200), it follows thatPm = |KT(wT 1)]1=t(216)soPPm= Pr;m (217)can be calculated.Middle (right): Equation (215) has been applied.Interpretation of Adsorption Isotherms 43Bottom (left):Asr(Y) =

1YwTwT Yt dY (218)The integration has been performed numerically.Bottom (right): The values of Eqs. (218), (212), and (217) have been conjugated.FIG. 19 The uniform and thermodynamically consistent interpretation of the Toth isotherms of Type I.The values of parameters are as follows: KL = 0:15 kPa1, wL = 1:05, t = 1:0, Pm = 133:3 kPa (solid line,mL equation); KT = 1:5 kPa1, wT = 1:05, t = 0:6, Pm = 75:0 kPa ( ); KT = 8 103kPa1,wT = 1:05, t = 1:6, Pm = 133:0 kPa ( ).44 To thIn Figure 19, the To th equations are represented with different parameters t (0.6, 1.0, 1.6). It isevident that t = 1 represents the modied Langmuir equation. The values t < 1 relate toheterogeneous surfaces where the adsorbentabsorptive interactions are greater than thosebetween the molecules adsorbed. The values t > 1 relate to the reverse situation. From thesesuppositions, it follows that the mL equation cannot be applied to homogeneous surfaces only,but it can describe isotherms measured on heterogeneous surfaces on the condition that the twointeractions adsorbent-adsorptive interactions and those between adsorbed molecules areapproximately equal. This suppositions implicit include the fact that the parameter t in theToth equation expresses not only the heterogeneity of the surface but also reects the interactionsand mobility (immobility) of the molecules adsorbed. This statement is proven in Section IV.F.It is also remarkable that the isotherms of Type I with different parameters t are verysimilar relationships; however, the corresponding functions c(P) in the top (right) of Fig. 19 arevery different and characteristic. Thus, the selection of the appropriate parameter t can be madewith the help of function (214).B. The Modied FowlerGuggenheim Equation Applied to HeterogeneousSurfaces (FT Equation)The modied, therefore, the consistent, mFG equation is applicable to homogeneous surfaces. Ashas been demonstrated, its explicit form isP = 1KmFYwF Y exp(BFY) (219)The corresponding function cmF(Y) has the formcmF(Y) = wFwF YBFY: (220)According to Toths idea Eq. (220) expresses the heterogeneity of the surface if the parameter as apower is introduced; that is,cFT(Y) = wFwF YtBFYtt > 0 (221)Substituting Eq. (221) into Eq. (95) and integrating, we obtainP = Pm(wF 1)1=texp BFt Y(wF Yt)1=t exp

BFYtt

(222)where the constant of integration has the formIFT = Pm(wF 1)1=texp BFt (223)Thus, we have the modied FG equation applicable for heterogeneous surfaces:P = 1KFT 1=tY(wF Yt)1=t exp BFYtt (224)whereKFT = (IFT)t= |Ptm(wF 1) exp(BF)]1(225)Interpretation of Adsorption Isotherms 45orPm = IFT (wF 1)1=texp BFt 1(226)From Eq. (224), as it is also valid that the Henry constant can be expressedlimP0YP= (KFTwF)1=t= H (227)Taking Eq. (206) into account, we haveKFT =ktBwFexp tUdiff0 (Y = 0)RT (228)that is, according to Eq. (209), KFT = KT.It is evident that Eq. (224) is a thermodynamically consistent equation becauselimY=1P = Pm (229)The types of isotherms described by Eq. (224) are equal to those described by the FG and mFGequations. The principle of calculations of the limiting values of BF and wF at which the differenttypes of isotherm can occur are also identical to those applied by the mFG equation in SectionIII.D. This is the reason why only the results are summarized here. In this sense, the value of BFwhere Eq. (132) is met isBF = wFYt(wF Yt) (230)The values of coverages where minima of Eq. (230) occur areYmin = (0:5wF)1=t(231)The corresponding minima of values of BF are unchanged [see Eq. (135)]:BF;min = 4wF(232)The values of BF where condition c(Y) = 1 is fullled areBF = 1wF Yt (233)Taking Eqs. (230)(233) into account, it can be veried that the limiting values of BF and wFdetermining the types of isotherms do not change. These values are shown in Fig. 8. Takingthese limiting values into account the different types of isotherm can be uniformly interpreted, asit has been done in Figs. 1012. In Fig. 20, the functions Y(P), zc(P), Y(Pr;m), C(Y), Asr(Y),and Asr(Pr;m) corresponding to isotherms of Type I are shown only because these functions occurmore frequently in practice than other types.The calculations relating to Fig. 20 are summarized as follows:Top (left):P = 1KFT 1=tY(wF Yt)1=t exp BFYtt (234)46 To thTop (right):cFT(Y) = wFwF YtBYt(235)The corresponding pairs (cFT; P) of Eqs. (234) and (235) are plotted.FIG. 20 The uniform and thermodynamically consistent interpretation of the FT isotherms of Type I. Thevalues of parameters are as follows: KmF = 0:10 kPa1, wF = 1:03, BF = 0:5, t = 1:0, Pm = 291:5 kPa(solid line, mF equation); KFT = 0:45 kPa1, wF = 1:03, BF = 0:90, t = 0:6, Pm = 202:2 kPa ( );KFT = 4 103kPa1, wF = 1:05, BF = 0:80, t = 1:6, Pm = 124:4 kPa ( ).Interpretation of Adsorption Isotherms 47Middle (left):Pr;m =Eq: (224)Eq: (226): (236)Middle (right): Equation (235) has been applied.Bottom (left):Asr(Y) =

lYcFT(Y) dY (237)The integration has been performed numerically.Bottom (right): The conjugated pairs of Asr(Y) and Eq. (236) are plotted.For practical applications of Eq. (4.29), a four-parameter tting process is proposed, wherethe parameters to be iterated are IFT, wT, t, and BF. IFT is dened by Eq. (223), Pm is calculablefrom Eq. (226), and KFT is determined by Eq. (225). The calculation of the total monolayercapacity, nsm, is discussed in detail in Section VI.C. The Modied Volmer Equation Applied to Heterogeneous Surfaces(VT Equation)The thermodynamically consistent