Solution Interfaces: A Theoretical

16514

J. Phys. Chem. B 2006, 110, 16514-16525

Kinetics of Solute Adsorption at Solid/Solution Interfaces: A Theoretical Development of the Empirical Pseudo-First and Pseudo-Second Order Kinetic Rate Equations, Based on Applying the Statistical Rate Theory of Interfacial Transport Wladyslaw Rudzinski*,†,‡ and Wojciech Plazinski‡ Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Cracow, Poland, and Department of Theoretical Chemistry, Faculty of Chemistry, UMCS, pl. M. Curie-Sklodowskiej 3, 20-031 Lublin, Poland ReceiVed: March 22, 2006; In Final Form: May 10, 2006

For practical applications of solid/solution adsorption processes, the kinetics of these processes is at least as much essential as their features at equilibrium. Meanwhile, the general understanding of this kinetics and its corresponding theoretical description are far behind the understanding and the level of theoretical interpretation of adsorption equilibria in these systems. The Lagergren empirical equation proposed at the end of 19th century to describe the kinetics of solute sorption at the solid/solution interfaces has been the most widely used kinetic equation until now. This equation has also been called the pseudo-first order kinetic equation because it was intuitively associated with the model of one-site occupancy adsorption kinetics governed by the rate of surface reaction. More recently, its generalization for the two-sites-occupancy adsorption was proposed and called the pseudo-second-order kinetic equation. However, the general use and the wide applicability of these empirical equations during more than one century have not resulted in a corresponding fundamental search for their theoretical origin. Here the first theoretical development of these equations is proposed, based on applying the new fundamental approach to kinetics of interfacial transport called the Statistical Rate Theory. It is shown that these empirical equations are simplified forms of a more general equation developed here, for the case when the adsorption kinetics is governed by the rate of surface reactions. The features of that general equation are shown by presenting exhaustive model investigations, and the applicability of that equation is tested by presenting a quantitative analysis of some experimental data reported in the literature.

Introduction The description of adsorption kinetics is a much more complicated problem than the theoretical description of adsorption equilibria. This is because the expressions describing the thermodynamic quantities at equilibrium are the only limiting forms of the expressions describing time evolution of these quantities under nonequilibrium conditions. Substantial progress has been made during the last few decades where gas/solid systems are concerned. The physical nature of solid/solution systems is much more complicated, which also makes the theoretical interpretation of the adsorption kinetics in these systems much more difficult. Meanwhile, the solid/solution systems are the adsorption systems of crucial importance for life on our planet and for a variety of important technological processes. It is very impressive to observe a real increase in the numbers of papers on the kinetics of sorption in the studies of environmental protection. So, it is also very intriguing to observe that the empirical kinetic equation proposed by Lagergren in 18981 is still “the most widely used rate equation for sorption of a solute from a liquid solution”:2

dNt ) k1(N(e) - Nt) dt

(1)

where Nt is the amount of the solute adsorbed at a time t, * Corresponding author; fax:+48-81-537-5685; e-mail: [email protected]. † Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences. ‡ Department of Theoretical Chemistry, Faculty of Chemistry, UMCS.

whereas k1 and N(e) are some constants. They can be easily found from the following linear regression of experimental data:

ln(N(e) - Nt) ) lnN(e) - k1t

(2)

which represents the integral form of eq 1, corresponding to the boundary condition Nt(t ) 0) ) 0. According to the commonly accepted interpretation, N(e) should be the amount adsorbed at equilibrium. This makes eq 1 a “pseudo-first order” equation, by comparison with the kinetic adsorption term of the “true” first-order equation in which N(e) is the maximum amount that can be adsorbed. The common use of the empirical Lagergren eq 1 led also to revealing deviations in many cases from the behavior predicted by eq 2. Some attempts were made to modify this equation by replacing the surface concentrations N(e) and Nt by the corresponding concentrations in the bulk solution.2 However, there was a general feeling that the origin of eq 1 is to be related to the kinetic model in which reaction on the surface, i.e., the transition from free to adsorbed state, is the mechanism controlling the rate of the adsorption process.2 By comparison with the adsorption term of the true first-order process, the form of eq 1 seemed to suggest a one-site-occupancy adsorption when the adsorbing molecule reacts with one adsorption site. Thus, it seemed natural to propose that in the case of two-siteoccupancy adsorption, i.e., when the solute molecule reacts with two adsorption sites, the rate of adsorption should be given by the following:

10.1021/jp061779n CCC: $33.50 © 2006 American Chemical Society Published on Web 08/02/2006

Solute Adsorption at Solid/Solution Interfaces

J. Phys. Chem. B, Vol. 110, No. 33, 2006 16515

dNt ) k2(N(e) - Nt)2 dt

(3)

Equation 3 has commonly been called the “pseudo-second-order rate equation”3. The integral form of this equation, obtained with the boundary condition Nt(t ) 0) ) 0, can be written in the following form:

t 1 1 ) t+ Nt N(e) k2(N(e))2

(4)

which has commonly been applied in the analysis of experimental data. It has been shown4-6 that, in many cases, eq 4 yields a better correlation of experimental data than eq 3. However, it should be emphasized that both eq 1 and eq 3 are essentially empirical equations. Thus, in view of the large applicability of eq 1 and its common use, during more than one century, one may wonder why no attempts were made to clearly explain its theoretical origin on the ground of the new fundamental theories of adsorption/desorption kinetics. The importance of the kinetics of adsorption in the solid/solution systems suggests a very urgent necessity of explaining this intriguing problem. Such explanations should lead us to a much better understanding of the mechanism and kinetics of adsorption in these systems. In their review on the problem in 2000, Ho et al. expressed the opinion that eq 1 “is only an approximate solution to the first-order rate mechanism”.2 Their review focused on the kinetics of pollutants adsorption, showing that this kinetics may be controlled by a variety of processes, like the transport to a solid surface, diffusion into pores, and other kinetic steps. Depending on the assumed rate controlling step, a variety of kinetic equations have been proposed in the literature. While discussing eq 1, Ho et al. classified this equation as corresponding to the case when the reaction on the surface is the rate controlling step. They also assumed that eq 1 must be a simplified form of another rate equation. While following that point of view, we have searched for a more general rate equation having a well-established theoretical background. Then, our efforts went toward applying this to the Statistical Rate Theory of Interfacial Transport (SRT). THEORY The Fundamentals of the SRT Approach to the Adsorption Kinetics. Assuming that the transport of molecules between two neighboring phases through their phase boundary results primarily from single molecular events, Ward and co-workers developed the expression for the rate of molecular transport R12 between two phases “1” and “2”. Using the first-order perturbation analysis and the Boltzmann definition of entropy they arrived at the following fundamental expression for R12:7

[ (

R12 ) Re exp

)

(

)]

µ1 - µ2 µ2 - µ1 - exp kT kT

(5)

Here µ1 and µ2 are the chemical potentials of the molecules in phases “1” and “2” at nonequilibrium conditions, and Re is the exchange rate at equilibrium to which the system would evolve after being closed and equilibrated. The theoretical grounds of that new fundamental approach were published in 1982.8 Soon it was applied to successfully describe the rates of interfacial transports of various kinds, i.e., the rate of exchange at the liquid/gas interface,7,9-11 hydrogen adsorption by metals,12 electron exchange between ionic isotopes in solution,13 permeation of ionic channels in biological membranes,14 and the rate of liquid evaporation.15-17 Ward and co-workers also first

showed how the SRT approach can be applied to describe the kinetics of isothermal adsorption at the gas/solid interfaces,18-21 and the kinetics of thermodesorption.22 Then a series of papers were published during the past decade by Rudzinski and coworkers, showing how that new description can be generalized further to describe the kinetics of isothermal gas adsorption/ desorption on/from energetically heterogeneous surfaces23-27 as well as the kinetics of thermodesorption.28-31 The new theory of adsorption/desorption kinetics successfully described some fundamental features of adsorption/desorption kinetics, observed in real gas/solid adsorption systems that could hardly be explained by the classical kinetics. For instance that, depending on the conditions under which kinetic experiment is carried out, the kinetic process may be described by a variety of kinetic equations.32 Very recently, the new SRT approach has been successfully applied to describe the kinetics of proton adsorption at the oxide/ electrolyte interfaces.33 It was proved for the first time that this adsorption kinetics is the source of the hysteresis frequently observed in both the classical potentiometric, and the calorimetric titration experiments. This leads us to assume that the new SRT approach may also explain the intriguing successful application of the empirical Lagergren equation for describing adsorption kinetics at the solid/solution interfaces. As the Lagergren kinetic equation has always been associated with the model of one-site-occupancy adsorption, we have accepted the Langmuir adsorption model in our theoretical investigation. Thus, we assumed that the chemical potential of the solute molecules adsorbed on a solid surface µs can be expressed as follows:

θ µs ) kTln - kTlnqs 1-θ

(6)

where θ ) Nt/Nm, and qs is the molecular partition function of the adsorbed solute molecule. In eq 6, Nm is treated as the maximum amount that can be adsorbed, usually called the adsorption capacity. In terms of the Langmuir model, this is the total number of the adsorption sites on the surface, available for adsorption of solute molecules. For the chemical potential of the solute molecules in the bulk solution µb, the following expression was assumed:

µb ) µb° + kTlnc

(7)

where c is the bulk solution concentration. Equations 6 and 7 lead to the Langmuir adsorption isotherm at equilibrium:

θ ) (e)

KLc(e)

(8)

1 + KLc(e)

where

( )

KL ) qs exp

µb° kT

(9)

and the superscript (e) will always denote equilibrium conditions. Now let us consider the corresponding SRT expression for the adsorption kinetics:

[ (

)

(

)]

µb - µs µs - µb dθ - exp ) K′ls exp dt kT kT

(10)

in which K′ls is the rate of adsorption at equilibrium. In the

16516 J. Phys. Chem. B, Vol. 110, No. 33, 2006

Rudzinski and Plazinski

Langmuir model of adsorption, K′ls is proportional to the frequency of the collisions of the solute molecules with the surface and to the number of the free molecules available for the adsorption sites (1 - θ(e)). Thus, we will write K′ls in the following form:

K′ls ) Klsc(e)(1 - θ(e))

(11)

While assuming that the rate of desorption is proportional to θ(e), we again arrive at the Langmuir isotherm (eq 8). Now, let us consider the classical kinetic equation of the fundamental TAAD approach (Theory of Activated Adsorption/Desorption), used throughout the last 20th century,

dθ ) Kac(1 - θ)s - Kdθs dt

(12)

[

]

(13)

Apparently, eq 13 would suggest an infinite rate of adsorption in the limit θ f 0. So, let us notice that the “thermodynamic” SRT approach does not apply when the rate predicted by eq 13 is higher than the flux of the solute molecules to surface. In that initial region purely kinetic arguments are to be considered like in the case of gas adsorption on solids. There, that problem was analyzed by Ward et al.8. Usually, it should be a small region of very low surface coverages. We postpone considering the special features of the kinetics in that region to our future publications. . Kinetics of Adsorption on Strongly Heterogeneous Surfaces Characterized by the Rectangular Adsorption Energy Distribution. While continuing our search for the source of the applicability of the Lagergren equation we have focused our attention on the following two facts that can be seen in the recently published literature reviews.2 (1) Most of the studies of the adsorption kinetics has been done by applying solids with evidently heterogeneous solid surfaces. (2) In many cases the authors showed a good applicability of the Elovich-type expression for the adsorption kinetics. In a series of papers we published recently on the kinetics of gas adsorption on solids, we have demonstrated that the Elovichtype expressions are obtained when the adsorption follows the Langmuir model of adsorption and the solid surface is strongly heterogeneous.34,35 Then, θ(e) is to be considered as a function of the energy of adsorption , related to qs:

qs ) qs° exp

(kT )

∫

l

(kT ) χ()d exp( ) kT

Kc(e) exp 1 + Kc(e)

(15)

where KL ) K exp(/kT) and χ() is the adsorption energy distribution normalized to unity. To arrive at the Elovich-type expressions for the kinetics of gas adsorption on solids, one has to assume a broad dispersion of adsorption energies, described by the following rectangular function:36

{

1 , for l e e m χ() ) m - l 0, elsewhere

(16)

In the case of solute adsorption considered, eqs 15-16 yield:

where s is the number of sites occupied by one adsorbed molecule, and Ka and Kd are temperature-dependent constants. From eq 12 one can arrive either at Lagergren (s ) 1), or at pseudo-second-order equation (s ) 2) by neglecting the desorption term in that fundamental TAAD equation, and yet defining θ as an “efficient” surface coverage equal to Nt/N(e). Now, let us see that the SRT expression (eq 10) does not lead to the Lagergren equation. This is because from eqs 6-10 we have the following:

θ 1-θ dθ 1 ) Klsc(e)(1 - θ(e)) KLc dt θ KLc 1 - θ

θt(e) )

m

(14)

The experimentally monitored total adsorption coverage θt(e) is then represented by the following average:

θt(e) )

kT Kc(e)em/kT + 1 ln (e) /kT m - l Kc e l + 1

(17)

One essential assumption inhered in the SRT approach is the assumption of a quasi-equilibrium on the solid surface. Namely, it is assumed that all correlation functions in the adsorbed phase are the same functions of surface coverage θt which one would observe at full equilibrium and the same surface coverage. Thus, the chemical potential of the adsorbed molecules µs should be the same function of surface coverage θt as under full equilibrium conditions. In the Langmuir model of adsorption one may consider the whole adsorption system as a collection of subsystems being only in thermal and material contact. Each subsystem corresponds to the sites having identical adsorption energy . Thus, the kinetics of adsorption on these sites should be expressed as follows:

[ (

)

µb - µs(θt) dθ() ) Klsc(e)(1 - θ(e)(,c(e))) exp dt kT µs(θt) - µb exp kT

(

)]

(18)

Then, by integrating both sides of eq 18 with χ() we obtain:

[ (

)

dθt µb - µs(θt) ) Klsc(e)(1 - θt(e)(c(e))) exp dt kT µs(θt) - µb exp kT

(

)]

(19)

At equilibrium

µs ) µb ) µb° + kTlnc(e)(θt(e))

(20)

c(e)(θt(e)) can be calculated from eq 17, which is the exact result of integration in eq 15, when χ() is the rectangular adsorption energy distribution (16). However, as in the case of gas adsorption on solids, we may apply the condensation approximation (CA),35,36 to arrive at an approximate, but also much simpler, result:

θt(e) )

m kT lnKc(e) + m - l m - l

From eqs 20 and 21 we have:

(21)


c(e) )

(


)

θt(e)(m - l) - m 1 exp K kT

dθt 2Klsc(e)(1 - θt(e))(m - l) (e) ) (θt - θt) dt kT

(22)

Thus, the expression µb - µs in eq 10 is now given by:

µb(c) - µs(θt) ) kTlnc + kTlnK + m - θt(m - l) (23)

We can see now that eq 27 is essentially identical with the Lagergren eq 1, and the theoretical interpretation of the coefficient k1 is following:

Therefore,

[

(

)

k1 )

-θt(m - l) + m dθt ) Klsc(e)(1 - θt(e)) Kc exp dt kT θt(m - l) - m 1 exp Kc kT

(

)]

(24)

Now, let us consider the fact that the function θt(t) monitored in a kinetic experiment will depend on the conditions under which this experiment is carried out. Most of the kinetic experiments is carried out in the following way: A certain amount of solution of an initial concentration c(in) is brought into contact with a certain known amount of solid adsorbent, usually in an Erlenmeyer flask. Next, the flask is shaken for a certain time ,and then the changing concentration of the solute is monitored. From the change of solute concentration, the amount adsorbed during the time is calculated. The experiments show that the coefficients k1 and k2 in the empirical eqs 2 and 4 depend, for instance, on the initial concentration c(in) but there is no theoretical explanation for that. Our SRT equation explains it for the first time because c(e) and θt(e) must depend on the amount of solid adsorbent, the volume of the solute solution, and its initial concentration. We faced a similar problem in the studies of gas adsorption kinetics. There, we studied, for instance the case of “volume dominated” systems, when the amount of the bulk molecules dominates the amount of adsorbed molecules in experiment to such an extent that the bulk concentration is essentially unchanged during the kinetic experiment. In our case, it means that the concentration c in eq 24 can be identified with the equilibrium concentration c(e). Then eq 24 can be rewritten to the following form:

[ ( (

) )]

t(θt) )

kT ln 2(m - l)K′ls

) ( ( ) ( )

)

(θt(e) - θt)(m - l) (θt(e) - θt)(m - l) - exp ) kT kT 3 2(m - l) (e) 1 m - l 1 (θt - θt) + (θt(e) - θt)3 + kT 3 kT 60 5 m - l (θt(e) - θt)5 + ... (26) kT

We can now observe the very interesting fact that the second term of this Taylor expansion disappears, while the terms of order [(θt(e) - θt)3] may be much smaller and, therefore, neglected. Then, from eqs 25-26 we have

(28)

{[ ( [ ( {[ ( [ (

) ] ) ]} ) ] ) ]}

θt(e)(m - l) exp +1 kT (θt(e) - θt)(m - l) exp -1 kT

θt(e)(m - l) -1 kT (θt(e) - θt)(m - l) exp +1 kT

The applicability of the Lagergren eq 1 would suggest that dθt/ dt is proportional to the difference (θt(e) - θt). So, let us expand the exponents within the square bracket into Taylor series around θt ) θt(e):

(

2Klsc(e)(1 - θt(e))(m - l) kT

Equations 27-28 suggest that the initial slope of the function θt(t) when t f 0 should increase with roughly second power of θt(e). As the value of θt(e) should increase with c(in), it also means that the initial slope of the function θt(t) should increase with the initial concentration of the solute in a kinetic experiment. Indeed, such an increase can be seen in many of the experimental data reported in the literature.2 It should, however, be emphasized that in the cases where the next term, 1/3((m - l)/kT)3(θt(e) - θt)3, in the Taylor expansion (eq 26) cannot be neglected, the dependence of dθt/ dt on (θt(e) - θt) may be a hybrid between the first and the third power dependences on (θt(e) - θt). Thus, it is to be expected that in such cases the assumed dependence on the second power of (θt(e) - θt) in eq 3 might simulate such a hybrid behavior. This would explain the theoretical background of the pseudo-second-order kinetic, eq 4, so frequently used to correlate experimental kinetic data. Our theoretical development of eq 27 would suggest that N(e) in the Lagergren empirical eq 1 should be identified with the maximum adsorbed amount, i.e., with the surface capacity. Some attempts to treat it as a free (adjustable) parameter should probably be seen as an empirical way to increase the applicability of Lagergren equation, being only a simplified form of the more exact kinetic eq 25. As the applicability of the Lagergren kinetic equation is examined by applying its integral form (eq 2), it is also worth emphasizing that the solution of the more exact kinetic eq 25 takes a simple compact form,

(θt(e) - θt)(m - l) dθt (e) (e) (e) ) Klsc (1 - θt (c )) exp - exp dt kT (θt(e) - θt)(m - l) (25) kT

exp

(27)

exp -

(29)

which can also be rewritten as follows:

θt ) θt(e) -

{ (

)

2(m - l)K′lst 2kT arctanh exp tanh m - l kT

[

]}

θt(e)(m - l) 2kT

(29a)

Our eq 24 also explains the applicability of the empirical Elovich equation, frequently reported in the literature. When the experiment is carried out in such a way that c ≈ c(e), and the second desorption term within the square bracket in eq 10 can



be neglected, eq 24 becomes, essentially, an Elovich-type kinetic equation:35

(

)

dθt Kls (e) (e) - θt(m - l) + m ) c θt exp dt KL kT

(30)

The integration with the boundary condition θt(t ) 0) ) 0, yields the following:

θt(t) )

[

]

Klsc(e)θ(e)(m - l) exp(m/kT) kT ln t+1 m - l KLkT

c)

(

) (

)

where V is the volume of the bulk solution. When c must be treated as a function of θt(t), the kinetic eq 24 must be numerically solved. Let us remark, however, that departures from the volume dominated character of the experiment will not affect the applicability of Elovich equation at small adsorption times (coverages) when the desorption term can be neglected. This is because the function c(in)(1 - Nm/N(in)θt(t)) can then be treated as the following Taylor expansion limited to the first two terms:

(

c ) c(in) 1 -

Nm

θ (t) (in) t

N

)

(

≈ c(in) exp -

Nm N(in)

)

θt(t)

(32)

With such approximation, eq 24 still preserves the form of the Elovich type differential equation

(

)

- θt(m - l) + m dθt Kls (in) (e) Nm ) c θt exp - (in)θt(t) dt KL kT N

]

[

]

(θt(e) - θt)(m - l) c(e) exp (34) c kT where τ is the dimensionless time:

Nt(t) N Nt(t) Nm N ) 1 - (in) ) c(in) 1 - (in)θt(t) V V V N N (31) (in)

[

dθt (θt(e) - θt)(m - l) c ) (e) exp dτ c kT

(31)

How the Experimental Conditions Affect the Monitored Kinetics of Adsorption. Thus, our SRT approach shows that both the empirical Lagergren and Elovich kinetic equations should apply best in the case of “volume dominated” kinetic experiments. When noticeable changes of the bulk concentration will be observed during a kinetic experiment, then deviations from the pseudo-first-order Lagergren equation are to be expected. In the case of the Elovich-type equation, additional deviations are to be expected due to neglecting the desorption term in eq 24. Our SRT approach also shows that simultaneous studies of adsorption equilibria are essential for a good understanding of adsorption kinetics. Let us state for this purpose that all the parameters in the developed kinetic equations but one are those appearing in the corresponding theoretical expression for the equilibrium adsorption isotherm. Thus, having determined these parameters by an appropriate quantitative analysis of experimental equilibrium adsorption isotherms, one should be able to predict the kinetic isotherms by adjusting just one more purely kinetic parameter. Then, the only unknown parameter in the kinetic eq 19 is the purely kinetic parameter Kls. When, in a kinetic experiment, noticeable changes in the bulk solute concentration are observed, c in eq 24 is to be treated as a function of θt(t), (in)

Now, let us consider again the full form of the kinetic eq 24, taking into account both the desorption rate and the change of the bulk concentration c in the course of adsorption. We may write it in the following form:

(33)

but the parameters kT/(m - l) and kT/mlnK estimated from the equilibrium adsorption isotherm may differ from those determined from kinetic data.

τ ) K′lst ) Klsc(e)(1 - θt(e))t

(35)

and c/c(e) is given by the expression:

c c(e)

c(in) )

Nmθt V

Nmθt(e) c(in) V

(36)

Then, for the purpose of our model investigations, we will write eq 34 in the following form:

dθt ) dτ λ - θt

exp[γ(θt(e) - θt)] (e)

λ - θt

λ - θt(e) exp[-γ(θt(e) - θt)] λ - θt (37)

where:

γ)

m - l kT

(38)

c(in)V Nm

(39)

and

λ)

Discussion of Theory and Experiment In view of the common use of the empirical pseudo-first and pseudo-second-order kinetic equations, we have decided to investigate the sources of their applicability or inapplicability in some cases. For that purpose we have made an extensive model investigation based on eq 37. In this equation, γ and λ are the parameters characterizing the physical features of an adsorption system and experimental conditions, respectively. From the adsorption literature treating on gas adsorption equilibria,35-37 one may deduce that typical values of the heterogeneity parameter γ may vary between 2 and 4. Then, the growing value of the technical parameter λ means that our adsorption system tends to be volume dominated. The relatively noticeable scatter seen in typical kinetic experiments on adsorption at the solid/liquid interfaces was probably the reason that the applicability of the pseudo-order kinetic equations was not studied by applying the differential forms of these equations. That applicability was studied by applying the integral forms of these equations to correlate the directly measured functions Nt vs t.


Figure 1. Derivative dθt/dτ calculated from eq 37 for the three values of the heterogeneity parameter γ, and the case of a volume dominated system, when λ - θt(e) ) 100.


Figure 4. Derivatives dθt/dτ shown in Figure 2, plotted now as the functions of (θt(e) - θt)2.

Figure 2. Derivative dθt/dτ calculated from eq 37 for the three values of the parameter λ - θt(e), when γ ) 3.

Figure 5. (A) Kinetic isotherms θt(τ/τ0.9) and (B) their Lagergren plots (2), calculated from eq 37, for θt(e) ) 0.5, λ ) 100, and the three values of the heterogeneity parameter: γ ) 2 (_____), γ ) 3 (- - - -), γ ) 4 (......).

Figure 3. Derivatives dθt/dτ shown in Figure 1, plotted now as the functions of (θt(e) - θt)2.

However, proper understanding of fundamental features of that adsorption kinetics requires studying the behavior of the derivative dθt/dτ first. Figure 1 shows that behavior in the case of typically volume dominated systems for various values of the surface heterogeneity parameter γ. Figure 2 shows how the technical parameter λ affects the behavior of dθt/dτ. Looking at Figures 1 and 2, we observe a poor linearity rather in the plots dθt/dτ vs (θt(e) - θt). This would suggest a poor applicability of the pseudo-first-order kinetic eq 1. So, in a next step we plotted the calculated functions dθt/dτ as the function of (θt(e) - θt)2 to see, whether the pseudo-second-order kinetic eq 3 represents better the calculated functions dθt/dτ. Figures 3 and 4 seem to suggest that, indeed, the empirical pseudo-second-order kinetic eq 3 represents better the behavior of the kinetic function dθt/dτ. However, this representation shows some serious deviations from linearity in some cases. This would suggest a generally limited applicability of both the pseudo-first and the pseudo-second order kinetic equations. Now let us see whether the commonly performed analyses of experimental data might lead to such a conclusion. These analyses have been based on applying the integral forms of these empirical equations. Let us see whether these integral forms can serve as a linear representation for the functions θt(t)

calculated from eq 37. We have to realize that the θt(t) functions, calculated from eq 37 for different sets of parameters, correspond to the behavior observed in various experiments. So we have to take into account the following essential fact. Namely, adsorption times monitored in different experiments may differ by several orders of magnitude, depending on the physical nature of an adsorption system. Thus, we have decided to study the behavior of θt as a function of the reduced time τ/τ0.9, where τ0.9 is the time when the surface coverage θt reaches the value θt ) 0.9θt(e). With such a choice of the reduced time, we have plotted in Figures 5-8 the functions ln(θt(e) - θt) vs τ/τ0.9 which, according to eq 2, should be linear when dθt/dτ follows the behavior predicted by the pseudo-first-order kinetic eq 1. Looking at Figures 5 and 6, one may see a fairly good linearity in these plots, except perhaps for the systems showing an extreme departure from a volume dominated system. This is especially true for surface coverages not close to the maximum possible surface coverage θt(e) ) 1. Figures 5-8 suggest that for many adsorption systems the linear integral plots ln(θt(e) - θt) vs t may be fairly linear functions, thus suggesting a good applicability of the pseudofirst order kinetic equation. However, as these plots are only apparently linear, their numerical linear regression may yield tangent and abscissa values, which will depend on both the initial concentration (λ values) and on the time intervals for which


Figure 6. (A) Kinetic isotherms θt(τ/τ0.9) and (B) their Lagergren plots (2), calculated from eq 37, for θt(e) ) 0.5, γ ) 3 and the three values of the technical parameter: λ ) 0.51 (_____), λ ) 1 (- - - -), λ ) 100 (......).

Figure 7. (A) Kinetic isotherms θt(τ/τ0.9) and (B) their Lagergren plots (2), calculated from eq 37, for θt(e) ) 0.9, λ ) 100, and the three values of the heterogeneity parameter: γ ) 2 (_____), γ ) 3 (- - - -), γ ) 4 (......).

the linear regression was made. It means that applying the linear regression (eq 2) to correlate the experimental data may lead to N(e) and k1 values, which will depend on experimental conditions, i.e., on the initial concentration and on the monitored times. Such dependence is, in fact, reported in many papers. One will face a similar situation when the integral form ( eq 4) of the pseudo-second kinetic eq 3 is applied to correlate kinetic data. Figures 9-12 show the behavior of (τ/τ0.9)/θt vs τ/τ0.9 functions calculated from our kinetic eq 37 for the same set of parameters, which were accepted in our model calculations presented in Figures 5-8. As the linearity of these plots is essential for the applicability of the pseudo-second-order kinetic eq 4, the parts (B) of these figures show the changes in the tangent of these plots. Looking at Figures 9-12, one can see fairly linear plots suggesting, thus, a good applicability of the pseudo-second-


Figure 8. (A) Kinetic isotherms θt(τ/τ0.9) and (B) their Lagergren plots (2), calculated from eq 37, for θt(e) ) 0.9, γ ) 3, and the three values of the technical parameter: λ ) 0.91 (_____), λ ) 2 (- - - -), λ ) 100 (......).

Figure 9. (A) Pseudo-second-order plot (eq 4), calculated from eq 37 for θt(e) ) 0.5, λ ) 100, and the three values of the heterogeneity parameter γ. (B) The tangents of the plots, drawn in panel A.

order kinetic eq 3. This is especially true for higher adsorption times. The linear regression of the plots presented in Figures 9-12 will lead one to N(e) and k2 parameters, which will depend on the initial solute concentration and on the interval of considered (monitored) adsorption times. Let us investigate now the meaning of the tangent and the abscissa of such linear plots on the ground of our kinetic eq 37. While studying the asymptotical behavior of the function θt(t), calculated from eq 37, one can show that

t/θt(t) 1 1 ) lim ) (e) tf∞ tf∞ θ (t) t θ t

lim

(40)

t

Thus, the interpretation of the tangent of the linear plot (4) which follows from our SRT approach is the same as that accepted so



Figure 10. (A) Pseudo-second-order plot (eq 4), calculated from eq 37, for θt(e) ) 0.5, γ ) 3, and the three values of technical parameter λ. (B) The tangents of the plots, drawn in panel A.

Figure 12. (A) The pseudo-second-order plot (eq 4), calculated from eq 37, for θt(e) ) 0.9, γ ) 3, and the three values of technical parameter λ. (B) The tangents of the plots, drawn in panel A.

Figure 11. (A) Pseudo-second-order plot (eq 4), calculated from eq 37 for θt(e) ) 0.9, λ ) 100, and the three values of the heterogeneity parameter γ. (B) The tangents of the plots, drawn in panel A.

theoretical values predicted by eq 41. This is because the linearity of the plots (τ/τ0.9)/θt vs (τ/τ0.9) soon becomes poor in the initial region of small adsorption times (τ/τ0.9). This creates a high level of uncertainty in a quantitative analysis of the abscissa determined from the experimental plots (eq 4). While looking for an experimental illustration of the features of the t/θt(t) vs t functions predicted by our theoretical approach, one faces the following problems: (1) The reported kinetic data in the literature are presented in a graphical form as a rule. This makes their quantitative analysis difficult due to a limited accuracy of both the experiment and its graphical presentation. (2) The kinetic experiments are rarely accompanied by measurements of equilibrium adsorption isotherms. This, of course, creates limitations in a theoretical quantitative interpretation. So, despite numerous data, which have already been published, it is not easy to select a set of data suitable for a quantitative theoretical analysis. First of all, the kinetics governed by surface reaction is only one of the possible kinetics, as it was shown in the review by McKay and co-workers.3 The selection of the proper kinetic model may not be easy, and the authors proposed several procedures which might be helpful for making such a selection. While applying such procedures they came to the conclusion that sorption of some dyes onto peat and wood best follows the pseudo-second-order kinetic model.38 Their paper reporting on sorption of two dyes, BB69 and AB25, onto peat and wood38 is similar to the papers where not only tabulated kinetic data were reported but also measurements of the related equilibrium isotherms. The four experimental equilibrium isotherms were analyzed using the Langmuir equation, but their fairly exact graphical presentation makes their digitizing sufficiently accurate for their analysis still based on other adsorption models. The graphically presented plots t/Nt(t) vs t for one of these systems (BB69 onto peat) seem to be fairly linear, and the tables in their paper reported the values of k2 and N(e) parameters determined from all the four plots. There is also one important table no. 5 in their paper comparing of the N(e) values determined from the kinetic plots (eq 4) with those determined from the equilibrium adsorption

far. Then, one can show that in the limit t f 0 the theoretical interpretation of the abscissa is the following:

[(

)]

λ - θt -γθt(e) t λ e lim eγθt(e) ) K′ls (e) tf0 θ (t) λ λ - θt t (e)

-1

(41)

In the limit of volume dominated systems, i.e., when λ f ∞, the expression (eq 41) takes the following simpler form:

t 1 lim ) tf0 θ (t) 2K′lssinh (γθt(e)) t

(42)

Equation 41 shows that the value of the abscissa will depend on both the physical features of an adsorption system, and on the technical conditions under which the kinetic experiment is carried out. However, Figures 5-12 also suggest that the values of the abscissa determined from the experimental integral plots (eq 4) may sometimes be dramatically different from the


Figure 13. Results of best fitting the measured equilibrium isotherms (0, O) by eq 17. The solid lines are the theoretical values calculated from eq 17, by accepting the following values of parameters. For peat: NmkT/(m - l) ) 2.6172mg,

Kem/kT ) 1.2264dm3/mg, Kel/kT ) 0.02065dm3/mg.

For wood: NmkT/(m - l) ) 2.6883mg,


Figure 15. Kinetics of BB69 adsorption on peat, studied at the two solute initial concentrations: c(in) ) 50 mg/dm3 and c(in) ) 100 mg/ dm3. The theoretical values (_____) were calculated from eq 37 by using the parameters determined from the equilibrium adsorption isotherm (17) and by assuming that Kls ) 2 × 10-4min-1. The experimental data points (O,4) were generated from eq 4 by using the k2 and N(e) values tabulated in the paper by Ho and McKay.38

Kem/kT ) 0.21635dm3/mg, Kel/kT ) 0.01367dm3/mg.

Figure 14. Comparison of the N(e)(c(in)) values found in kinetic experiments (0, O) with the theoretical values (_____) calculated from eq 17 for the same two sets of parameters as those listed in Figure 13. Other parameters were taken from the description of the experiments in the paper by Ho and McKay38.

isotherms. Here the best agreement can be observed for BB69 adsorbed on peat, somewhat worse for BB69 adsorbed on wood, and dramatic differences (up to 30%) in the case of AB25 adsorbed on both peat and wood. So, we started our quantitative analysis based on eq 37 by considering the kinetics and equilibria of BB69 adsorbed on peat and wood. Thus we fitted the digitized data points for the equilibrium adsorption isotherms by the exact eq 17. Figure 13 shows the agreement between theory and experiment, along with the values of the best-fit parameters. Having determined the equilibrium isotherm parameters, we have drawn, in Figure 14, these isotherms as the functions of the initial concentration c(in) by considering the technical conditions of the experiment (mass of adsorbent, volume of solution, and its concentration). The N(e)(c(in)) values calculated in this way are compared in Figure 14 with the N(e)(c(in)) data points determined by Ho and McKay in their kinetic experiment. In the case of BB69 adsorbed on peat, a good agreement is observed for the first three lowest initial concentrations, c(in), but a noticeable deviation of the last kinetic data point corresponding to the highest initial concentration c(in) ) 500 mg/dm3 is observed. In the case of BB69 adsorbed on wood, a small systematic deviation can be observed for all the three kinetic data points. Having determined the parameters of the equilibrium adsorption isotherm (eq 17), we used them to fit the experimental kinetic plots t/θt vs t by adjusting the same value of the

Figure 16. Kinetics of BB69 adsorption on peat, studied at the two solute initial concentrations: c(in) ) 200 mg/dm3 and c(in) ) 500 mg/ dm3. The theoretical values (_____) were calculated from eq 37 by using the parameters determined from the equilibrium adsorption isotherm (17) and by assuming that Kls ) 2 × 10-4min-1. The experimental data points (O, 4) were generated from eq 4 by using the k2 and N(e) values tabulated in the paper by Ho and McKay.38 The black circles (b) in Figure 16 are the values generated by eq 4 by assuming that the N(e) value is the one predicted by our theoretical fit of the equilibrium adsorption isotherm.

parameter Kls for all initial solute concentrations. As the original data points Nt(t) were not tabulated by the authors, we simply numerically generated their best linear regression (eq 4) by using the parameters k2 and N(e) tabulated by Ho and McKay in their paper. While treating these generated plots as true experimental ones, we fitted them by using the θt(t) values calculated from eq 37, by accepting the parameters determined from the equilibrium adsorption isotherm (eq 17) and a certain best-fit value of Kls. The results of these calculations are shown in Figures 15-17. Looking at Figures 15 and 16 one can see a surprisingly good agreement between our theory and experiment for the three lowest initial concentrations, and a worse agreement for the highest concentration c(in) ) 500 mg/dm3. This deviation in reproducing kinetic experiments must obviously be related to the worse estimation of N(e) in the kinetic experiment for this initial concentration, as it could be seen in Figure 14. The worse agreement between theory and experiment for the highest initial concentration c(in) ) 500 mg/dm3 seen in Figure 16, can be improved by replacing the value N(e) found in the kinetic experiment by the value of N(e) determined from the equilibrium adsorption isotherm. As in the case of AB25 adsorption on peat and wood, the differences between these two values become very large (up to 30%), we have given up on a quantitative analysis of that adsorption kinetics, based on applying our eq 37 to the data generated by the linear regression (eq 4).



Figure 17. Kinetics of BB69 adsorption on wood studied at three solute initial concentrations: c(in) ) 50 mg/dm3, c(in) ) 100 mg/dm3, and c(in) ) 200 mg/dm3. The theoretical values (_____) were calculated from eq 37 by using the parameters determined from the equilibrium adsorption isotherm (17) and by assuming that Kls ) 2 × 10-4min-1. The experimental data points (],0,4) were generated from eq 4 by using the k2 and N(e) values tabulated in the paper by Ho and McKay.38

At the same time, we applied this procedure to BB69 adsorbed on wood, for whose system the reported differences seem to be small, as can be seen in Figure 14. The results of these calculations are shown in Figure 17. One interesting feature of the results presented in Figures 15-17 is that the estimated Kls value is almost the same for BB69 adsorbed on both peat and wood. This would suggest that the kinetic constant Kls is mainly related to the features of solute molecules, but at present we do not have a clear explanation for that. As the pseudo-experimental data generated from eq 4 are linear functions, we could not check whether our theoretical functions might not fit better the original data θt(t) found in that kinetic experiment. However, we may provide an impressive proof that the trends observed in the behavior of our theoretical functions t/θt(t), presented in Figures 9-12, really reproduce the trends observed in the behavior of experimentally measured data. Let us consider, for instance, the t/Nt(t) data reported by Ho and McKay,4 for copper (II) ions adsorbed onto a commercial granular activated carbon at the three initial concentrations. The dependence of t/Nt(t) on t was seen by Ho and McKay as sufficiently linear for the linear regression (eq 4) to be applied. Indeed, from a first view, the dependence on time of the t/Nt(t) experimental data may look to be fairly linear, as shown in Figure 18A. However, a more detailed insight into the graphically presented data reveals some systematic deviations of the measured t/Nt(t) data from their linear regression (eq 4). A similar behavior can be seen in the case of sorption kinetics of Omega Chrome Red ME on fly ash, as shown in Figure 18B. Our model calculations presented in Figures 9-12 show, that the deviations from the linear behavior of the theoretical t/θt(t) functions can be more clearly seen in the calculated derivatives of these functions. Thus, we have decided to check whether the deviations from a linear behavior of the t/Nt(t) vs t data reported by these authors may not have the character predicted by our model calculations based on our eq 37. For that purpose, we digitized the graphically reported t/Nt(t) vs t data, and approximated them numerically by a function smoothing these data locally. Next, we calculated the derivative of that smoothing function. The results of our exercises are shown in Figures 19 and 20. Looking at Figures 19 and 20, we can see that the behavior of the derivatives determined from the experimentally measured functions t/Nt(t) is the same as that predicted by our model calculations presented in Figures 9-12. The derivative is a decreasing function at small adsorption times, and passes next through a minimum to reach a constant value at high adsorption

Figure 18. (A) t/Nt(t) experimental data for copper (II) adsorbed onto commercial granular activated carbon (GAC) measured by Ho and McKay at the three initial concentrations: c(in) ) 10 mg/dm3 (0), c(in) ) 15 mg/dm3 (O), and c(in) ) 20 mg/dm3 (]). The data (0,O,]) were digitized from Figure 12 of their paper and the solid lines are their linear regression; tmax ) 540 min is the time for which the maximum adsorbed amount was recorded. (B) The kinetic data for the Omega Chrome Red ME adsorption on fly ash measured at the three initial concentrations: c(in) ) 5 mg/dm3 (9), c(in) ) 10 m g/dm3 (b), and c(in) ) 15 mg/dm3 ([). The data (9,b,[) were digitized from Figure 15 of their paper and the solid lines are the corresponding linear regressions; tmax ) 100 min.

Figure 19. (A) Experimental data points t/Nt for copper (II) ions adsorbed onto a commercial granular activated carbon reported by Ho and McKay for the initial concentration c(in) ) 15 mg/dm3 (0). The data (0) were digitized from Figure 12 of the paper by Ho and McKay, and next approximated by a function (______) smoothing these data. Here tmax ) 540 min is the time for which the maximum adsorbed amount was recorded. (B) The derivative (______) (tangent) of the function (______) smoothing the experimental data in panel A.

times. The model calculations presented in Figures 9-12 suggest that in some cases only a decreasing tendency may be observed. Such a tendency can, for instance, be observed in the experimental plots t/Nt(t) reported by Ho and McKay for Omega Chrome Red E, adsorbed on fly ash. Having not known the data for equilibrium adsorption isotherms, we could carry out only such quantitative analysis


Figure 20. (A) Experimental data points t/Nt for Omega Chrome Red ME adsorbed onto fly ash, reported by Ho and McKay for the initial concentration c(in) ) 10 mg/dm3 (9). The data (9) were digitized from Figure 15 of the paper by Ho and McKay, and next approximated by a function (______) smoothing these data. The meaning of tmax ) 100 min and of the solid lines (______) here is the same as in Figure 19.

of the reported experimental plots t/Nt(t). Fitting them by eq 37 would involve the necessity of numerical determination of four best-fit parameters, which would make such numerical exercises risky. Thus, again, we would like to emphasize that carrying out simultaneous studies of adsorption equilibria is essential for a proper understanding of adsorption kinetics. The observed deviations from the linear plots (eqs 2 or 4) were sometimes interpreted as switching from one kind of kinetics to another. Of course, applying other criteria like the effect of agitation speed or effect of particle size is also essential for understanding the adsorption kinetics, as discussed in the review by Ho et al.2. Nevertheless, using a proper theoretical expression to describe some kind of kinetics seems to be crucial for proper interpretation of the observed kinetics. In this paper we limited our interest to the adsorption kinetics, which is governed by the rate of surface reactions. Our studies confirm the common interpretation that the pseudo-first and the pseudosecond order kinetic equations should be associated with that kind of kinetics, but they are only simplified forms of a more general kinetic equation. One possible form of such general equation has been developed here by applying the new approach to the rate of interfacial transport, called the Statistical Rate Theory. Summary Kinetics of solute adsorption at the solid/liquid interfaces is the essential feature of these systems for their applications in technology and environmental protection. Great progress has been made during the past decades in the theoretical description of adsorption equilibria in these systems. On the contrary, relatively little progress has been made in the description of adsorption kinetics. The Lagergren empirical equation launched at the end of 19th century is still the most widely used equation to correlate experimental data on adsorption kinetics. Meanwhile, no attempts were made to explain the theoretical origin of this equation on the ground of the new fundamental theories of adsorption/desorption kinetics. One faces the same situation in the case of the very popular kinetic equation called the

Rudzinski and Plazinski pseudo-second order rate equation. That equation is just an intuitive generalization of the Lagergren equation, also called frequently the pseudo-first order rate equation. These equations related the rate of adsorption to the adsorbed amount. A general feeling was expressed that these empirical equations are simplified forms of a more general equation describing the rate of adsorption in the cases when it is governed by the rate of surface reaction. While following that general feeling we have decided to search for the theoretical origin of these equations on the ground of the new fundamental theory of interfacial transport called the Statistical Rate Theory (SRT). Then, looking at the numerous papers reporting on the successful applications of the pseudo-first and pseudo-second order kinetic equations, we made the following observation. These were always adsorption systems with highly heterogeneous solid surfaces, usually low-cost adsorbents used to purify polluted effluents containing dyes/organics or metal ions. Thus, we assumed a model of a strongly energetic heterogeneous solid surface, along with the commonly accepted Langmuir model of adsorption of solute molecules. Application of the SRT fundamental approach also makes it possible to take into consideration the technical conditions under which the kinetic experiment is carried out (mass of adsorbent, volume of solution, and its initial concentration). In this way, we arrived at a differential equation for the rate of adsorption which reproduces very well all the essential features of the observed adsorption kinetics. It is shown that for certain values of physical and technical parameters, our new general equation reduces to the famous Lagergren empirical equation. It is also shown that our new general equation may also explain the behavior of adsorption kinetics predicted by the empirical pseudo-second-order rate equation. Model investigations based on our new general kinetic equations show that deviations of the observed kinetics from the behavior predicted by the pseudofirst and the pseudo-second-order kinetic equations may be due to the approximate character of these equations. However, we also demonstrate that proper interpretation of the observed kinetics requires carrying out a simultaneous study of adsorption equilibria. References and Notes (1) Lagergren, S. Kungliga SVenska Vetenskapsakademiens. Handlingar 1898, 24 (4), 1. (2) Ho, Y. S.; Ng, J. C. Y.; McKay, G. Sep. Purifi. Methods, 2000, 29 (2), 189. (3) Ho, Y. S.; McKay, G. Water Res. 2000, 34 (3), 735. (4) Ho, Y. S.; McKay, G. Process Biochem. 1999, 34, 451. (5) Sharma, A.; Bhattacharyya, K. G.; J. Hazard. Mater. 2005, B125, 102. (6) Vijayaraghavan, K.; Jegan, J. R.; Palanivelu, K.; Velan, M. Electron. Jo. Biotechnol. 2004, 7 (1), 61. (7) Ward, C. A. J. Chem. Phys. 1977, 67, 229. (8) Ward, C. A.; Findlay, R. D.; Rizk, M., J. Chem. Phys. 1982, 76, 5599. (9) Ward, C. A.; Rizk, M.; Tucker, A. S., J. Chem. Phys. 1982, 76, 5606. (10) Ward, C. A.; Tikuisis, P.; Tucker, A. S., J. Colloid Interface Sci. 1986, 113, 388. (11) Tikuisis P.; Ward, C. A. In Transport Processes in Bubbles, Drops and Particles; Chabra, R., DeKee, D., Eds.; Hemisphere: New York, 1992; p 114. (12) Ward, C. A.; Farabakhsk, B.; Venter, R. D. Z. Phys. Chemie 1986, 147, 8. (13) Ward, C. A. J. Chem. Phys. 1983, 79, 5605. (14) Skinner, F. K.; Ward, C. A.; Bardakjian, B. L., Biophys. Chem. 1993, 65, 618. (15) Ward, C. A.; Fang, G. Phys. ReV. E 1999, 59, 429. (16) Fang, G.; Ward, C. A. Phys. ReV. E 1999, 59, 441. (17) Fang, G.; Ward, C. A., Phys. ReV. E, 1999, 59, 417. (18) Ward, C. A.; Elmoseli, M., Surf. Sci. 1986, 176, 457. (19) Elliott, J. A. W.; Ward, C. A., Langmuir 1997, 13, 951.

Solute Adsorption at Solid/Solution Interfaces (20) Elliott, J. A. W.; Ward, C. A. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (21) Elliott, J. A. W.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (22) Elliott, J. A. W.; Ward, C. A. J. Chem. Phys. 1997, 106, 5677. (23) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (24) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2001, 105, 6858. (25) Panczyk, T.; Rudzinski, W. J. Phys. Chem. B 2002, 106, 7846. (26) Rudzinski, W.; Panczyk, T. Langmuir 2002, 18, 439. (27) Panczyk, T.; Rudzinski, W. Langmuir 2003, 19, 1173. (28) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1997, 13, 3445. (29) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1999, 15, 6386. (30) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. J. Phys. Chem. B 2000, 104, 1984.

J. Phys. Chem. B, Vol. 110, No. 33, 2006 16525 (31) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir 2000, 16, 8037. (32) Rudzinski, W.; Panczyk, T. Adsorption 2002, 8, 23. (33) Piasecki, W. Langmuir 2003, 19, 9526. (34) Rudzinski, W.; Panczyk, T. Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalizations of the Elovich Equation. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, C., Eds.; Marcel Dekker: New York, 1999. (35) Rudzinski, W.; Panczyk, T. J. Non-Equilib. Thermodyn. 2002, 27, 149. (36) Rudzinski W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (37) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (38) Ho, Y. S.; McKay, G. Trans IChemE, 1998, 76 B, 313.

Solution Interfaces: A Theoretical

Recommend Documents