adsorption theory, modeling, and analysis

All consequences are physically and thermodynamically nonsense; however, in spite of this fact,the functions and isotherms having these contradictions can be applied excellently in pract

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This book presents some apparent divergences, that is, its content branches off in manydirections This fact is reflected 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 ramified research fields, have similar or identicalroots These statements mean that this book is 1) diverse, but still unified and 2) classical, butstill modern The book contains monographs at a scientific 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, Fowler–Guggenheim, 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
Y appears as denominator in any of the mentioned relationships, then in the limiting case

The mathematical and thermodynamical consequences of these facts are the following:

1 The monolayer adsorption can be completed only when the equilibrium pressure isinfinitely great

2 The change in thermodynamic functions are also infinitely great when the monolayercapacity is completed

3 The adsorption potential tends to infinity when p tends to zero

iii

All 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 applyLangmuir’s and=or Pola´nyi’s 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 book—explicitly or implicitly—the question of thermodynamic consistency or inconsistency emerges,and the first 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 1–9 deal mostly with gas–solidadsorption and Chapters 10–15 deal with liquid–solid adsorption Chapter 2 discusses the gas–solid 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 4–7 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 liquid–solidadsorption discussed in Chapters 10–15 can be regarded as developments and=or continuations

of Everett’s and Shay’s 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 gas–solid and liquid–solid 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

Jo´zsef To´th

Salil U Rege and Ralph T Yang

4 Adsorption Isotherms for the Supercritical Region 211

Li Zhou

5 Irreversible Adsorption of Particles 251

Zbigniew Adamczyk

6 Multicomponent Adsorption: Principles and Models 375

Alexander A Shapiro and Erling H Stenby

7 Rare-Gas Adsorption 433

Angel Mulero and Francisco Cuadros

8 Ab Fine Problems in Physical Chemistry and the Analysis of Adsorption–

Desorption Kinetics 509

Gianfranco Cerofolini

9 Stochastic Modeling of Adsorption Kinetics 537

Seung-Mok Lee

10 Adsorption from Liquid Mixtures on Solid Surfaces 573

Imre De´ka´ny and Ferenc Berger

v

11 Surface Complexation Models of Adsorption: A Critical Survey in the Context

of Experimental Data 631

Johannes Lu¨tzenkirchen

12 Adsorption from Electrolyte Solutions 711

Etelka Tomba´cz

13 Polymer Adsorption at Solid Surfaces 743

Vladimir Nikolajevich Kislenko

14 Modeling of Protein Adsorption Equilibrium at Hydrophobic Solid–Water

Interfaces 803

Kamal Al-Malah

15 Protein Adsorption Kinetics 847

Kamal Al-Malah and Hasan Abdellatif Hasan Mousa

Index 871

Zbigniew Adamczyk Institute of Catalysis and Surface Chemistry, Polish Academy ofSciences, Cracow, Poland

Kamal Al-Malah Department of Chemical Engineering, Jordan University of Science andTechnology, Irbid, Jordan

Ferenc Berger Department of Colloid Chemistry, University of Szeged, Szeged, HungaryMalgorzata Boro´wko Department for the Modelling of Physico-Chemical Processes, MariaCurie-Sklodowska University, Lublin, Poland

Gianfranco 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, Ukraine

Seung-Mok Lee Department of Environmental Engineering, Kwandong University,Yangyang, Korea

Johannes Lu¨tzenkirchen Institut fu¨r Nukleare Entsorgung, Forschungszentrum Karlsruhe,Karlsruhe, Germany

Hasan Abdellatif Hasan Mousa Department of Chemical Engineering, Jordan University

of Science and Technology, Irbid, Jordan

Angel Mulero Departmento de Fisica, Universidad de Extremadura, Badajoz, Spain

vii

Salil U Rege* Department of Chemical Engineering, University of Michigan, Ann Arbor,Michigan

Alexander A Shapiro Department of Chemical Engineering, Technical University ofDenmark, Lyngby, Denmark

Erling H Stenby Department of Chemical Engineering, Technical University of Denmark,Lyngby, Denmark

Etelka Tomba´cz Department of Colloid Chemistry, University of Szeged, Szeged, HungaryJo´zsef To´th Research Institute of Applied Chemistry, University of Miskolc, Miskolc-Egyetemva´ros, Hungary

Ralph T Yang Department of Chemical Engineering, University of Michigan, Ann Arbor,Michigan

Li Zhou Chemical Engineering Research Center, Tianjin University, Tianjin, China

*Current affiliation: Praxair, Inc., Tonawanda, New York.

Uniform and Thermodynamically

Consistent Interpretation of

Adsorption Isotherms

JO´ ZSEF TO´TH Research Institute of Applied Chemistry, University of Miskolc,

Miskolc-Egyetemva´ros, Hungary

I FUNDAMENTAL THERMODYNAMICS OF PHYSICAL ADSORPTION

A The Main Goal of Thermodynamical Treatment

It is well known that in the literature there are more than 100 isotherm equations derived based

on various physical, mathematical, and experimental considerations These variances are justified

by 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 Isotherms

Let us suppose that a solute in a solution has surface tension g ðJ=m2Þ The value of g changes as

a 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, ns

1are the moles of solute and ns

2are those of the solution, the

1

free surface is Asðm2Þ, the chemical potentials are ms

dGs¼ ss

dT þvs

dP þ ms1 dns1þ ms

where ssis the entropy of the Gibbs phase (J=K) and vs

is its volume ðm3Þ Lete us compare Eqs.(2) and (3) so we get for constant values of As, T , and P;

Asdg þ ns1dms1ns2dms2¼ 0 ð4ÞThe same relationship can be applied to the bulk phase with the evident difference that here

Asdg ¼ 0

that is,

n1dm1þ n2dm2¼ 0 ð5Þwhere the symbols without superscript s refer to the bulk phase

For the sake of elimination, let us multiply dm2from Eq (4) by ns

2=n2and take into accountthat

2mol solution, whereas in the bulk phase ns

2ðn1=n2Þ mol solute is present with n2mole solution The difference between the two amounts is the total surface excess amount, ns

If the chemical potential of the solute material is expressed by its activity, that is,

as 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:

C Derivation of the Gibbs Equation for Adsorption on Liquid=Solid

Interfaces Adsorption Isotherms

The 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 ðgÞ, the free energy of the surface, As

ðJ=m2Þ, is introduced and applied because, evidently, g cannot be measured on S=Linterfaces From the thermodynamical point of view, there is no difference between As

and g

2 In several cases, the surfaces Asðm2Þ of solids cannot be exactly defined or measured.This statement is especially valid for microporous solids According to the IUPACrecommendation [1], in this case the monolayer equivalent area ðAs;eÞ determined bythe Brunauer–Emmett–Teller (BET) method (see Section VI) must be applied As;ewould result if the amount of adsorbate required to fill 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 dAsþ ns

1
ns 2

The basic idea of this method is the following Let the composition of a binary liquidmixture be defined by the mole fraction of component 1; that is,

x1;0¼ n1 ;0

n1;0þ n2;0¼

n1;0

where 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 again

be defined by the mole fraction of component 1:

x1¼ n1

n1þ n2¼

n1;0
ns 1

n1

n2



ð32ÞLet us write Eq (32) in the form:

Gs1 ¼ A
1

s



ns1
ns 2

da1

Equation (35) provides the possibility for calculating the change in free energy of the surface,

DAs

1, if the activities of component 1 are known In dilute solutions, a1 x1; therefore, in thiscase, the calculation of DAs

1 by Eq (35) is very simple

The most complicated problem is to calculate or determine the composite (absolute)isotherms ns

1versus x1and ns

2versus x2because, 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 relationship

ns1f1þ ns

where f1 and f2 are the areas effectively occupied by 1 mol of components 1 and 2 in themonolayer Gibbs phase ðm2=mol) From Eqs (36), (28), and (29), we obtain the compositeisotherms

ns1¼ Asx1þ f2nnðsÞ1

f1x1þ f2ð1
x1Þ ð37Þand

ns2¼Asð1
x1Þ
f1nnðsÞ1

f1x1þ f2ð1
x1Þ ð38ÞEquations (37) and (38) can be applied when—in addition to the monolayer thickness—thefollowing conditions are also fulfilled: (1) The differences between f1 and f2 are not greaterthan 30%, (2) the solution does not contain electrolytes, and (3) lateral and vertical interaction

do not take place between the components In Fig 1 can be seen the five types of isotherm, nnðsÞ1versus x1, classified for the first 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 defined 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=Solid

Interfaces

This 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 defined 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 first law of thermodynamics applied to a normal three-dimensional one-componentsystem is the following:

dU ¼ T dS
P dV þ m dn ð39Þwhere U is the internal energy (J), S is the entropy (J=K), V is the volume ðm3Þ, m is the chemicalpotential (J=mol), P is the pressure ðJ=m3Þ, and n is the amount of the component (mol)

FIG 1 The five types of excess isotherm nnðsÞ1 versus x1classified by Schay and Nagy [4].

FIG 2 The composite monolayer isotherms corresponding to the five types of excess isotherm andcalculated by Eqs (37) and (38).

Let 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,

nsdms¼
Ss

dT þ VsdP þ AsdAs ð42ÞDividing both sides of Eq (42) by ns, we have the chemical potential of the Gibbs phase:

On the right-hand side of Eq (46),vgnsis the volume ðm3Þ of ns in the bulk (gas) phaseand nsvs is the volume of ns in the Gibbs phase It means that the difference

Thus, Eq (48) can be written in the modified form

must be taken into account

E The Differential Adsorptive Potential

The 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., we

do not know how many and what kind of physical parameters or quantities influence the energy(heat) processes connected with the adsorption] As it is well known these heat processes can beexactly measured in a thermostat of approximately infinite 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 ðdT ¼ 0Þ or, by converse processes (desorption), the heat

is 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, an

amount of heat dQ (J) be absorbed by the thermostat at constant T Thus, the general definition

of the differential heat of absorption is

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 rolling

on the Earth’s 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 defined as follows Let us denote the rotational energy of 1 mol adsorptive as Urotg and

Us is that in the adsorbed (Gibbs) phase So, the change in the rotational energy is

v is the vibrational energy of 1 mol adsorbed molecules and Uv;0s is the vibrational

energy of those at 0 K If the above-mentioned potential energy is denoted by U0; then we obtain

qdiffh ¼ U0þ DUrþ DUt
DUs

vþ Us

where the subscript h refers to homogeneous surface and Us

l is the energy which can beattributed to the lateral interactions between molecules adsorbed Equation (63) can be written in

a shortened form if the two changes in kinetic energies are added:

The energy connected with the lateral interactions, Us

l, depends on the coverage (i.e., the greaterthe coverage or equilibrium pressure, the larger is Us

l 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 write

Wi¼ Dditið1
YiÞ ð66Þwhere Wiis the probability of finding a molecule adsorbed on the ith patch, Ddiis the extent ofthe patch (expressed as a fraction of the whole surface), t is the relative time of residence of the

molecule on the ith patch, and Yi is the coverage of the same patch In this sense, it can bedefined an average or differential adsorptive potential, formulated as follows:

Because the kinetic energies and Us

l do not change from patch to patch (i.e., they areindependent of U0;i), we can write

U0diff ¼ qdiff
Us

As will be demonstrated in the next section, the thermodynamic parameter functions, As and

U0diff 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 EQUATIONS

A The Basic Phenomenon of Inconsistency

In Section I.D., it has been proven that the exact Gibbs equation (48) contains the surface excessvolume, Vs, defined by the relationship

Vs¼ nsðvg
vsÞ ð72Þwhere nsis the measured adsorbed amount (mol) andvg

andvs

are the molar volume ðm3=molÞ

of the measured adsorbed amount in the gas and in the adsorbed phase, respectively Equation(72) means that nsshould be equal to the equation

37 MPa  P  139 MPaÞ, the compressibility factor Z varies approximately as a linear function:ZðpÞ ¼ 0:0682p þ 0:356 ð74Þ

Taking into account that

isotherms have maximum values

pressure, vg > vs

is valid The functions nsðPÞ (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 tends

to infinity ðP ! 1Þ, cannot exist

B Inconsistent G=S Isotherm Equations

In 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:

measured on GAC activated carbon at 298 K decreases approximately linearly (From Ref 8.)

In Eqs (80) and (81), ns

mis the total monolayer capacity, U0 is the constant adsorptive potential,and kB is defined by de Boer and Hobson [10]:

kB¼ 2:346ðMTÞ1 =2 105 ð82Þwhere 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,

In Section II.A, it has been proven that according to the Gibbs thermodynamics, a plateau

in the range of great pressure cannot exist; therefore, the Langmuir equation is 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

thermodynami-III THE UNIFORM AND THERMODYNAMICALLY CONSISTENT TWO-STEPINTERPRETATION OF G=S ISOTHERM EQUATIONS APPLIED FOR

HOMOGENEOUS SURFACES

The elimination of the thermodynamical inconsistency of the isotherm equations can be done intwo steps the first 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

A 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 suppositions

vg vs and the applicability of the ideal-gas law

Let us introduce in Eq (55) the coverage defined by Eq (80); we now obtain

AsðPÞ ¼ As

id

ðPmP

that is, jmis equal to the surface covered by 1 mol of adsorptive at Y ¼ 1 It is easy to see that

Eq (86) is the free energy of the surface when the total monolayer is completed ðns¼ ns

mÞ andthis monolayer behaves as an ideal two-dimensional gas Therefore, As

id can be applied as areference value; that is,

P Pm, the ideal-gas law is applicable and the suppositionvg  vs

is valid The applicability of

Eq (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 of

Eq (89) is supported by the fact that the suppositionvg vs in 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 Connection

Between the First and Second Steps

Let us introduce a differential expression having the form

The values of functions (90) and (91) also do not change when the adsorbed amounts areexpressed in coverages, Y:

cðYÞ ¼Y

P

dYdP


1

ð92ÞLet us write Eq (92) in this form:

dPcðPÞP

cðYÞ

Y dY

ð98Þand

ns

ns0¼ Y0¼ exp

ðP0P

dPcðPÞP

From Eq (100) follows the exact thermodynamical meaning of the function cðyÞ:

Eq (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 definite 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 fulfilled 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 defined 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 18

in Section III.H)

C The Uniform and Consistent Interpretation of the Modified Langmuir

Equation, General Considerations

Let us apply the implicit integral relationships (96) for derivation of the Langmuir equation Forthis reason, it must be demonstrated that the function cðPÞ belonging to the Langmuir equationis

In Eq (109), the condition that at P ¼ Pmnsis equal to the total monolayer capacity is fulfilled[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 infinity [i.e., the total monolayer capacity is completed at aninfinitely great equilibrium pressure ðwL¼ 1Þ].

Before demonstrating other properties of the mL equation, it is necessary to prove thevalidity of Eq (103); that is,

2 The function cðYÞ of the inconsistent equation cannot be applied for this tion [i.e., the integration of Eq (95) or (98) with the inconsistent functions cðyÞ doesnot lead to consistent isotherms equations] However, the inconsistent functions cðyÞare applicable to prove the inconsistency of the thermodynamical functions [see Eq.(112)]

transforma-3 The reason for statement (1) is the fact that the function cðYÞ has a concretethermodynamical meaning defined by Eq (102) Therefore, all thermodynamicalconsistencies or inconsistencies are directly reflected by the function cðYÞ

4 From statements (2) and (3), it follows that in Eqs (95), (98), and (100), only theconsistent form of the function cðYÞ can be applied

How these consistent forms of cðYÞ 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 defined by Eq (100) To calculate this change, the explicit form of the function cðYÞ

is required This function, belonging to the original Langmuir equation has the following form:

Equation (112) reflects a thermodynamic inconsistency because the change in free energy of thesurface never can be infinite However, if we substitute the function cmLðyÞ [Eq (109) into Eq.(100)], we have

DAsrðY ¼ 1Þ

ð1 0

Eq (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 value

lim

P!1 Y ¼ 1

which is thermodynamically inconsistent It is also proven that all inconsistencies are reflected

by the function cðYÞ 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 are

Y ¼ f ðPÞ However, most of the isotherm equations cannot emplicitly be expressed in this form,

as only the term P ¼ WðYÞ exists In these cases, the calculation of the function cðPÞ is thefollowing First, the function cðYÞ is calculated, which, in this case, has the form

cmLðYÞ ¼ wL

cmLðYÞ ¼ cmLðPÞ ð118Þ

to every value of P calculated by Eq (115) can be attributed a value of cmLðYÞ Thus, we obtainthe function cmLðPÞ numerically It is evident that if the isotherm investigated can be expressedboth in terms of Y ¼ f ðPÞ and P ¼ jðYÞ, then the two methods lead to the same relationship,cðPÞ

The practical importance of the function cðPÞ is discussed in Section III.A For Fig 6(middle, left), this interpretation of the measured isotherms is thermodynamically exact, because

if reflects the fact that the integration in Eq (97) has been performed to the finite upper limit Pm

and, therefore, the equilibrium pressure should be expressed in a relative pressure defined as

For Fig 6 (middle, right), the function cðYÞ 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)]:

of Pr;m¼ P=Pm calculated by Eqs (115) and (120) are attributed the values of AsðYÞ, so weobtain the functions AsðPr;mÞ These two types of function in the bottom of Fig 6 characterizethermodynamically the adsorption process and thus seem to complete the uniform interpretation

of the mL and other isotherm equations The thermodynamic consistency is best reflected by thefunctions AsðYÞ and AsðPr;mÞ because both functions have finite values at Y ¼ 1 or at Pr;m¼ 1

D The Uniform and Consistent Interpretation of the Modified

BF ¼Co

In Eq (123), o is defined 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 values

lim

P!o Y ¼ 1 and lim

y!1 P ¼ 1are also valid for Eq (122)

To obtain a consistent form of Eq (122), let us calculate its function cðyÞ:

cFðYÞ ¼ 1

It has been proven that the consistent Langmuir equation has the function cmLðYÞ defined by Eq.(109); therefore, the consistent (modified) Fowler–Guggenheim (mFG) equation should have thefollowing function:

IF ¼ PmðwF
1Þ expðBFÞ ð127ÞLet us compare Eqs (126) and (127) with Eq (122); we have

P ¼ 1

KmF

Y

wF
Y expð
BFYÞ ð128Þwhere

KmF¼ I
1

F ¼ ½PmðwF
1Þ expðBFÞ
1 ð129Þor

Pm¼ IF½ðwF
1Þ expðBFÞ
1 ð130ÞTherefore, the consistent form of the FG equation is relationship (128) For example, the limitingvalue of Eq (128) is

of isotherms are interpreted

From the physical meaning of the mFG relationships, it follows that they have to reflect thetwo-dimensional condensation too, similar to the three-dimensional van der Waals equation Inthis case, the isotherm equation,

P ¼ jðYÞ

should have local maximum and local minimum values These values of y exist when thecondition

by solid lines

In Fig 8, it can be seen also that functions BFðYÞ have absolute minimum values Afterdifferentiation of Eq (133), we have the values of coverage where these minima occur:

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 ;min¼ 4

In Fig 8, the values of BF ;minare represented by horizontal dotted lines.

For the determination of other types of isotherm corresponding to Eq (128), it is essential

to calculate the function BFðYÞ which fulfils the condition

According to Eq (125), condition (136) is met by the following values of BF and Y:

BF ¼ 1

In Fig 8, functions (137) are shown with dash-dot-dot lines

V, and condensation)

Mathematically, the condition cðYÞ ¼ 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 cðYÞ> 1 and cðYÞ < 1 and the point where the proportionalline drawn from the origin is a tangent can be seen Evidently, cðYÞ ¼ 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 first figure ðwF ¼ 1Þ 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 infinity when Y tends to 1 In this fact is alsoreflected the thermodynamical inconsistency of the original FG equation In the top (right) ofFig 8, wF ¼ 1:17, the place of minimum, Ymin, according to Eq (134) has been increased So,from the analysis and figures above, it follows that the Type I isotherm is described when

0< BF < 1=wF, Type V isotherms can occur when 1=wF < BF < 4=wF, and two-dimensionalcondensation takes place when B > 4=w A very interesting limiting case is shown at the

bottom (left) of Fig 8 when wF ¼ 1:333: In this case, the minimum value of BF ;min; that is, Eq.

(134) at Y ¼ 1 is equal to the value of BF corresponding to Eq (137):

Type I when 0< BF < 1=wF

Type V when 1=wF < BF < 1=ðwF
1Þ

Type III when 1=ðwF
1Þ < BF < 4=wF

Type condensation when BF > 4=wF

Thus, it is proven why the four types of isotherm shown in Fig 7 can be described by themodified FG equation

The analysis made above is limited to mathematical considerations In Figs 10–12 thethermodynamically consistent and uniform interpretation of isotherms Types I, III, and V,respectively, are shown In these figures, quite equal to Fig 6, are represented the functionsYðPÞ, cðPÞ, YðPr;mÞ, cðYÞ, AsðYÞ, and AsðPr;mÞ The calculations of these functions have beenmade as follows:

Top (left): The applied mFG equations are

P ¼ 1

KmF

Y

wF
Y expð
BFYÞ ð140ÞThe types of isotherms in Figs 10–12 have been determined by the corresponding values

of w and B (see Fig 8)

FIG 10 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type I.

FIG 11 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type III.

FIG 12 The uniform and thermodynamically consistent interpretation of the mFG isotherms of Type V.

Top (right): The functions cðPÞ by the relationships

cmFðYÞ ¼ wF

wF
Y
BFY ð141Þand

cmFðYÞ ¼ cmFðPÞ ð142ÞMiddle (left): The thermodynamically consistent and uniform interpretation of themeasured isotherms in form of YðPr;mÞ; that is

Middle right: The functions cðYÞ are bases for calculation of functions AsðYÞ and AsðPr;mÞ:Bottom: The functions AsðYÞ and AsðPr;mÞ are calculated numerically by the integralequation

AsðYÞ ¼ As

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 fitting procedure is recommended In particular, Eq (126) can bewritten in the form

P ¼ IF

Y

wF
Y expð
BFYÞ ð148ÞEquation (148) can be fitted to the measured points ðY; PÞ with parameters IF, wF, and BF Theaverage percentile deviation ðD%Þ has been calculated using the following relationship:

where P is the measured equilibrium pressure, Pcis calculated equilibrium pressures, and N isthe number of the measured points ðY; PÞ.

We have the constants IF, wF, and BF as results of the fitting procedure According to Eq.(129), we obtain

P!1 Y ¼ 1 and lim

Y!1 P ¼ 1are on contradiction with the Gibbs equation (51) and it is also unacceptable that according to

dYð1
YÞ2¼ 1 ð156ÞThese 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 modifiedVolmer (mV) equation can be written:

cmVðYÞ ¼ wV

w
Y

2

ð157Þ