15100
J. Phys. Chem. C 2007, 111, 15100-15110
Studies of the Kinetics of Solute Adsorption at Solid/Solution Interfaces: On the Possibility of Distinguishing between the Diffusional and the Surface Reaction Kinetic Models by Studying the Pseudo-First-order Kinetics 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, Maria Curie-Sklodowska UniVersity (UMCS), pl. Marii Curie-Sklodowskiej 3, 20-031 Lublin, Poland ReceiVed: April 27, 2007; In Final Form: August 9, 2007
It is shown that the popular pseudo-first-order Lagergren equation can be applied to correlate kinetic adsorption data only in the adsorption systems that are not far from equilibrium. Also, it is shown that the Lagergren equation is then the limiting form of the kinetic equations developed by assuming both diffusional and surface reaction kinetic models. However, the theoretical interpretation of the coefficients in the Lagergren equation is then different for the two different kinetic models. The comparison of the theoretically predicted values of these coefficients with their experimentally determined values creates a chance to conclude whether the diffusional or the surface reaction model should be assumed to represent the kinetics of adsorption in an investigated adsorption system.
Introduction Adsorption at a solid/solution interface is probably the most commonly applied physical process in the industrial purification of wastewaters. There, the sorbent and the solution are brought into contact for a limited period of time. So, not only knowing the equilibrium features of a solid/solution system is important, but knowledge of its kinetic features is also essential in supplying the fundamental information required for the design and operation of adsorption equipments for wastewater treatment. For plant simulations, simple relationships between the adsorption performance and the operating conditions are preferred. At the same time, understanding of the nature of a kinetic process is essential for proper design of operating conditions. So, theoretical model interpretations of the adsorption kinetics have been the subject of many publications, starting from the first ever theoretical paper published by Lagergren at the end of the 19th century.1 Then a variety of theoretical models has been proposed for modeling of that adsorption kinetics. The most popular of them may be classified as belonging to the following two groups: (1) the group of models assuming that the intraparticle diffusion controls the rate of adsorption, and (2) the group of the models where the rate of adsorption is assumed to be governed by the rate of the adsorption reactions occurring at a solid/liquid interface. While trying to recover the nature of a kinetic process from a theoretical analysis of experimental kinetic data, one faces the following two difficult problems: first, both the diffusional and the surface reaction kinetic models predict basically similar behavior of kinetic data for certain ranges of adsorption times and surface coverages; next, the theoretical origin of certain popular kinetic expressions is not well understood, so their applicability cannot serve as a key to understanding the nature of a kinetic process. * Corresponding author; Fax:+48-81-537-5685; e-mail: rudzinsk@ hermes.umcs.lublin.pl. † Polish Academy of Sciences. ‡ Maria Curie-Sklodowska University.
For instance, among the most popular kinetic equations are the so-called “pseudo-first-order” and “pseudo-second-order” rate expressions. Both of them assume that the difference between the actual and the equilibrium surface concentrations is the driving force for adsorption. The first of them assumes that the plot ln(N (e) - N(t)) should be a linear function of time (t). Here, N(t) is the actual adsorbed amount (surface concentration) and N (e) is the amount adsorbed at equilibrium, (i.e., N (e) ) N(t f ∞)). In this paper the superscript (e) will always denote equilibrium conditions. The pseudo-second-order kinetic expression assumes a linearity of the plot t/N(t) versus t, respectively.2,3 However, in spite of their common use, they have remained essentially empirical equations until very recently. In 2004, Azizian proposed their first interpretation based on the classical fundamental theory of activated adsorption/desorption approach (TAAD).4 Then, in 2006, Rudzinski and Plazinski proposed their first interpretation based on the new fundamental statistical rate theory (SRT) approach to adsorption/desorption kinetics.5 These theoretical interpretations show that both the pseudofirst- and the pseudo-second-order kinetic expressions are “lumped” forms of some more general expressions for adsorption kinetics. These lumped forms can be used if certain experimental conditions are fulfilled. For instance, it appears that the pseudo-first-order Lagergren equation should generally be applicable for the range of longer adsorption times, such that an adsorption system is not far from equilibrium. But then, the kinetic features predicted by the pseudo-first-order kinetic expression become very similar to those predicted by the diffusional model. We have analyzed it extensively in our recent publication.6 For longer adsorption times, the Lagergren equation appears to also be the lumped form of a popular, more general expression, developed for the diffusional kinetic model. Of course, one may face the situation that, even at longer adsorption times, the popular Lagergren equation does not apply. This would mean that either the experimental conditions are not proper for the lumped pseudo-first-order expression to be used or that the underlying physical model is not applicable.
10.1021/jp073249c CCC: $37.00 © 2007 American Chemical Society Published on Web 09/22/2007
Kinetics of Solute Adsorption at Solid/Solution Interfaces
J. Phys. Chem. C, Vol. 111, No. 41, 2007 15101
Quite often, the linear regression t/N(t) versus t, offered by the pseudo-second-order lumped expression, appears to be a more efficient way to correlate experimental data.3 However, the frequently observed better applicability of the linear plot t/N(t) versus t requires further theoretical studies. Is that better applicability due to the mathematical feature of such linear representation, or is it due to the physical nature of an adsorption system and the conditions under which the kinetic experiment is carried out? Such studies have already been started in two recent publications,6,7 but they seem to be in the stage where some deeper and more extensive theoretical studies are still needed. We plan to present such study in our future publication. Meanwhile, the purpose of the present publication is to carry out an exhaustive theoretical study of the origin and applicability of the pseudo-first-order kinetic equation. In particular, our theoretical analysis will focus on the possibility of distinguishing between the diffusional and the surface reaction kinetics, in the systems where the pseudo-first-order Lagergren expression describes well the behavior of the observed adsorption kinetics. Theory 1. The Intraparticle Diffusion Model and the Model of a Homogeneous Solid Surface. According to the intraparticle diffusion model (IDM), intraparticle diffusion is the only rate controlling step. The IDM can be derived from Fick’s second law, under certain additional assumptions.8 The most widely used mathematical expression corresponding to this model is eq 1,
N(t)
)1-
N (e)
6 π
∞
∑ 2 n)1
1 n2
(
Dn2π 2
exp -
r2
t
)
(1)
where D is the intraparticle diffusivity, and r is the particle radius. The form of eq 1 is not particularly useful for a simple analysis of experimental data. However, for certain experimental conditions, some approximations can be made. For short times, when the fractional uptake N(t)/N (e) is less than around 0.3, eq 1 practically reduces to eq 2.
N(t) N (e) At long times, when N(t) f form (eq 3).
N(t) N (e)
)1-
6 ) r
x
N (e),
eq 1 reduces to another simple
Dt π
(
6 Dπ 2 exp t π2 r2
eq 2 is often applied to correlate experimental data for the whole measured kinetic isotherm.10-12 Generally, the kinetics of solute sorption at solid/solution interfaces is less extensively studied in the range of low surface coverages (short adsorption times) because of the well-known experimental problems of monitoring very fast initial kinetics of adsorption. Thus, it seems more promising to use eq 3 to analyze experimental data in a wide range of adsorption times when an adsorption system is not far from equilibrium. Equation 3 can be rewritten to the following form (eq 4).
ln(N (e) - N(t)) ) -
(4)
By properly adjusting the value of N (e), one can make the lefthand-side of eq 4 a linear function of time. The tangent of such a linear plot is then equal to -Dπ2/r2, and its abscissa is equal to (ln N (e) + ln 6/π2). Of course, the value of the N (e) parameter determined from the abscissa should be equal to that which makes the plot ln(N (e) - N(t)) a linear function of time. It should be noted that both eqs 2 and 3 are only simplified (lumped) forms of eq 1 and can be used only in suitable limited ranges of surface coverages (adsorption times). The basic formula (eq 1) should be applicable to correlate experimental kinetic data recorded for both short and long adsorption times. However, fitting the experimental data for such wide regions of surface coverages (adsorption times) involves the problem of a proper definition of error function. Its choice may affect the conclusions drawn from such best-fit exercises.7 Therefore, the possibility of drawing such conclusions from a simple fitting procedure will always offer serious advantages in theoretical analysis of experimental kinetic data. 2. The Surface Reaction Model of Adsorption Kinetics and its Classical Interpretations. The Lagergren equation, also known as the pseudo-first-order equation, launched at the end of 19th century, is the earliest ever known equation describing the kinetics of adsorption. It is usually expressed as shown in eq 5,
dN(t) ) k1(N (e) - N(t)) dt
(2)
)
Dπ 2 6 t + ln N (e) + ln 2 2 r π
(5)
where k1 is a constant. Until very recently, eq 5 has been treated as an essentially empirical one. It has only very recently been shown by Azizian4 that the Lagergren equation may be derived from the first-order kinetic equation of the classical fundamental TAAD approach (eq 6), used throughout the last 20th century,
(3) dN(t) ) Kac(Nm - N(t)) - KdN(t) dt
According to eq 2, a plot of N(t) versus xt should be a straight
line with a slope equal to 6N (e)xD/r 2π and an intercept around zero. Such a linear dependence has, so far, commonly been treated as a test for whether that diffusion is the rate-controlling step. Meanwhile, we recently proved that the square root dependence on time of the monitored adsorption N(t) cannot be treated as definite proof that the rate of adsorption is controlled by the intraparticle diffusion process.6,9 While applying the new fundamental SRT approach to adsorption/ desorption kinetics, we have shown that such square-root dependence on time should also hold for the model where adsorption kinetics is controlled by the rate of surface reaction. Equation 2 is very frequently used to correlate experimental kinetic data. However, despite the fact that its applicability is limited to low surface coverages (short initial adsorption times),
(6)
where Ka and Kd are some constants, c is the solute concentration in the bulk phase, and Nm is the monolayer capacity. When dN(t)/dt ) 0 (i.e., when equilibrium is reached), eq 6 leads to the Langmuir adsorption isotherm (eq 7),
N
(e)
)
NmKLc(e) 1 + KLc(e)
(7)
in which
KL ) Ka/Kd
(8)
When accepting the additional assumption c ≈ c(e), eq 6 can be combined with eq 7 to yield the following expression (eq9),
15102 J. Phys. Chem. C, Vol. 111, No. 41, 2007
dN(t) ) (Kd + Kac(e))(N (e) - N(t)) dt
Rudzinski and Plazinski
(9)
which is identical with the Lagergren eq 5. Then, the theoretical interpretation of the coefficient k1 is as follows (eq 10).
k1 ) Kd + Kac(e)
(10)
The important assumption in the above derivation of the Lagergren equation is that solute concentration must be constant during the whole kinetic experiment (i.e., c is not a function of N(t)). Generally, there are two situations that one may face in the course of a kinetic experiment: (1) the adsorption system is the so-called “volume dominated” system. This is when the amount of the bulk molecules dominates the amount of the adsorbed molecules in the experiment to such an extent that the bulk concentration does not practically change during that kinetic experiment. It also means that the solute concentration c can be identified with the equilibrium concentration c(e) and with the initial concentration c(in). (2) In the course of a kinetic experiment, significant changes in the solute concentration are observed. When this occurs, the solute concentration c is the function of the adsorbed amount N(t). The adsorption system is the so-called “solid dominated” system when almost all of the adsorbate is removed from the solution during the experiment. Most of the kinetic experiments reported in literature were carried out in such a way that one had to take into account the change in the bulk concentration of solute (c) in the course of experiment. This would suggest a poor applicability of the Lagergren equation. However, when the adsorption system is close to equilibrium, one can assume that the solute concentration c becomes fairly constant and close to c(e). Thus, on the ground of the TAAD fundamental approach to adsorption/ desorption kinetics, this would suggest that, in most of the real kinetic experiments, the applicability of the Lagergren equation is limited to the range of higher adsorption times, where c f c(e) and N(t) f N (e). Generally, the value of the actual solute concentration is given by eq 11,
c ) c(in) -
N(t) V
(11)
where V is the volume of solution. Then, the combination of eqs 6 and 11 yields eq 12.
(
)
N(t) dN(t) (Nm - N(t)) - KdN(t) ) Ka c(in) dt V
(12)
It has been also shown by Azizian4 that, under certain conditions, eq 12 reduces to the other commonly used formula to describe the kinetics of adsorption: the pseudo-second-order equation. The Lagergren equation and the pseudo-second-order equation have usually been associated with the case when adsorption kinetics is governed by the rate of surface reaction. Of course, when assuming V f ∞ we have c ) c(in), and we face the case of a “volume dominated” system. The differential eq 5, when solved with the boundary condition N(t ) 0) ) 0, yields eq 13
N(t) ) N (e)(1 - e-k1t)
(13)
The following linear representation based on this equation has commonly been used in the analysis of experimental data (eq 14).
ln(N (e) - N(t)) ) ln N (e) - k1t
(14)
Thus, properly adjusting the value of N (e) in the left-hand-side of eq 14 should make ln(N (e) - N(t)) a linear function of time with the slope equal to - k1 and the abscissa equal to ln N (e). Note that just like in the case of the diffusional eq 4, the condition that must be fulfilled is the linearity of the ln(N (e) N(t)) versus t plot. However, there is an important difference between the theoretical interpretation of the abscissas in eq 4 (βD),
βD ) ln N (e) + ln
6 π2
(15)
and in eq 14 (βL).
βL ) ln N (e)
(16)
Let us note that the difference between them is constant and is equal to ln 6/π2. The fact that conditions for both eqs 4 and 14 are identical can be easily explained when we write the diffusional eq 3 in its differential form (eq 17).
dN(t) Dπ 2 (e) ) 2 (N - N(t)) dt r
(17)
We can now see that, like the case of the Lagergren equation (eq 5), the rate of adsorption is proportional to the difference (N (e) - N(t)), and the theoretical interpretation of the k1 coefficient is eq 18.
k1 )
Dπ 2 r2
(18)
To arrive at eq 3 we have to accept as the boundary condition the situation when, at t ) 0, some preadsorbed amount exists (i.e., N(t ) 0) ) N0). Then, after solving eq 17 and transforming the solution into the form useful for analysis of experimental data, we obtain eq 19,
ln(N (e) - N(t)) ) ln N (e) where we have eq 20.
t0 ) -
Dπ2
(
(t + t0)
(19)
)
(20)
r2
N0 r2 ln 1 - (e) 2 Dπ N
When assuming the boundary condition N0 ) N (e)(1 - 6/π2), we obtain the diffusional eq 3. For this case we have eq 21.
t0 ) -
r2 6 ln 2 2 Dπ π
(21)
One can observe that, for the systems where intraparticle diffusion governs the adsorption kinetics, the product t0Dπ2/r2 should be constant and equal to -ln 6/π2, independent of the initial solute concentration. In fact, the procedure based on introducing the additional best-fit parameter t0 and leading to improvement of the applicability of the Lagergren equation has already been proposed in literature13 and applied for correlation of measured kinetic isotherms. The simple analysis of the product t0Dπ2/r2 makes it possible to check if adding parameter t0 is equivalent to applying eq 3 instead.
Kinetics of Solute Adsorption at Solid/Solution Interfaces 3. The Model of Heterogeneous Solid Surface. So far, our theoretical interpretation of the kinetics of adsorption has been based on assuming the classical Langmuir adsorption model. This is a model of an ideal localized (monolayer) adsorption onto a lattice of energetically equivalent adsorption sites. Because the solute-solid interactions can be seen as relatively strong ones, the model of localized adsorption can hardly be questioned. On the other hand, the model of energetically equivalent adsorption sites can hardly be accepted for many adsorption systems applied in environmental protection. The large industrial scale of these adsorption processes involves the necessity of using sorbents that are made of cheap and widely available materials. Generally, these cheap, widely available materials are sorbents with highly irregular surfaces, characterized by strong geometric and energetic heterogeneities. So, it is no surprise that the measured equilibrium isotherms are not exactly Langmuir isotherms. They may look like Langmuir isotherms to a first, crude approximation, but a closer inspection usually shows that they cannot be correlated well by the Langmuir equation. So, a variety of expressions has been used in the environmental adsorption literature to correlate experimental data for adsorption equilibria. Thus, the frequently observed poor applicability of Langmuir equation again raises the intriguing question of the origin of Lagergren kinetic equation. In our recent theoretical paper, we showed that the Lagergren kinetic equation can also be derived by assuming the Langmuir model of adsorption on a strongly energetically heterogeneous solid surface.5 This could be done by applying the new fundamental approach to the adsorption kinetics (SRT) and accepting a rectangular adsorption energy distribution to represent the strong energetic heterogeneity of adsorption sites. However, in our previous development, the additional assumption made was that the adsorption system is volume dominated. Here, we are going to show that the Lagergren equation should also hold for the systems that are not strongly volume dominated but are not far from equilibrium. For that purpose, we briefly repeat here the generalization of the Langmuir equation for the energetically heterogeneous surface characterized by the rectangular adsorption energy distribution. For the chemical potential of the solute molecules in the bulk solution µb, the following expression can be assumed (eq 22),
µb ) µ°b + kT ln c
(22)
where the chemical potential of the solute molecules adsorbed onto a solid surface (µs), corresponding to the Langmuir model of adsorption, can be expressed as eq 23,
µs ) kT ln
θ - kT ln qs 1-θ
(23)
where qs is the molecular partition function of the adsorbed solute molecule, and θ is the surface coverage defined as θ ) N/Nm. When equilibrium is reached (µs ) µb) eqs 22 and 23 lead to the Langmuir adsorption isotherm (eq 7) with the coefficient KL defined as follows.
()
µ°b KL ) qs exp kT
(24)
Equation 23 is correct only when assuming the classical Langmuir model of adsorption on an energetically homogeneous lattice of adsorption sites. When the real, energetically heterogeneous, solid surfaces are considered, qs varies from one site
J. Phys. Chem. C, Vol. 111, No. 41, 2007 15103 to another and is characterized by an adsorption energy () by eq 25.
qs ) qs° exp
(kT )
(25)
Then, the total equilibrium surface coverage θ(e) t is given by the following average,
θ(e) t )
∫
l
(kT ) χ() d exp( ) kT
Kc(e) exp
m
1 + Kc(e)
(26)
where KL ) K exp(/kT), and χ() is the distribution of adsorption sites among the adsorption energy values (), normalized to unity. Further, l and m are the lowest (l) and the maximum (m) values of the adsorption energy () onto the heterogeneous lattice of adsorption sites. It is common in the adsorption literature to assume that when the solid surface is strongly heterogeneous the broad dispersion of adsorption energies can be described well by the following rectangular function χ().14-16
{
1 for l e em χ() ) m - l 0 elsewhere
(27)
From the combination of eqs 26 and 27, one obtains the following equilibrium isotherm equation (eq 28).
θ(e) t )
kT Kc(e)em/kT + 1 ln (e) /kT m - l Kc e l + 1
(28)
After applying the condensation approximation method,15-17 one may arrive at an approximate but also much simpler result (eq 29).
θ(e) t )
m kT ln Kc(e) + m - l m - l
(29)
In the next section we are going to show that accepting the rectangular adsorption energy distribution again leads to the Lagergren equation at longer adsorption times, when the surface reaction model is assumed. However, the interpretation of the equation coefficients is then different than in the case of the IDM approach. So, this may create a chance to discriminate between the diffusional and surface reaction kinetic models. 4. The Principles of SRT for the Kinetics of Solute Adsorption on Homogeneous Solid Surfaces. The theoretical grounds of the SRT of interfacial transport were published in 1982.18 That new approach was succesfully applied to describe the rates of interfacial transports of various kinds. Very recently, the SRT has been also used to describe the kinetics of solute adsorption at the solid/solution interfaces.5,6 The basic SRT expression for the adsorption kinetics is given by eq 30,
[ (
) (
)]
µb - µ s µs - µ b dθ - exp ) K′ls exp dt kT kT
(30)
where K′ls is the rate of adsorption at equilibrium, and µb and µa are the chemical potentials of the molecules in the bulk (b) and adsorbed (a) phases, under nonequilibrium conditions. With good approximation, the coefficient K′ls can be written as the following product (eq 31),
15104 J. Phys. Chem. C, Vol. 111, No. 41, 2007
Rudzinski and Plazinski
K′ls ) Klsc(e)(1 - N (e)/Nm) ) Klsc(e)(1 - θ(e))
(31)
because, in the case of the Langmuir model of adsorption, one may assume K′ls is proportional to the frequency of collisions of the solute molecules with the surface and to the number of free adsorption sites available for the adsorbing molecules (1 - θ(e)). For the classical Langmuir model of adsorption, the chemical potential of the solute molecules adsorbed onto a homogeneous solid surface (µs) can be expressed by eq 23. Then, from eqs 22, 23, and 30 we have eq 32,
[
θ 1-θ dθ 1 ) Klsc(e)(1 - θ(e)) KLc dt θ K Lc 1 - θ
]
(32)
in which KL is defined in eq 24. Because of changing solute concentration in the bulk phase, we must treat c in eq 32 as a function of the amount adsorbed. For that purpose we can rewrite eq 11 as eq 33.
c ) c(in) -
θNm N(t) ) c(in) V V
(33)
Then, eq 32 can be rewritten as eq 34,
dθ η(λ - θ)(1 - θ) θ ) dτ θ η(λ - θ)(1 - θ)
(34)
where η ) KLNm/V, λ ) c(in)V/Nm, and τ ) K′lst ) Klsc(e)(1 θ(e))t At equilibrium, when dθ/dτ ) 0, we arrive at Langmuir’s equilibrium adsorption isotherm (eq 35),
θ(e) )
1 + η + ηλ - x(1 + η + ηλ)2 - 4η2λ 2η
(35)
c(e)
in which the equilibrium solute concentration is expressed by the corresponding initial concentration c(in). When the equilibrium adsorption isotherm is expressed by eq 35, we can rewrite eq 34 as follows.
θ(λ - θ(e))(1 - θ(e)) θ(e)(λ - θ)(1 - θ) dθ ) dτ θ(λ - θ(e))(1 - θ(e)) θ(e)(λ - θ)(1- θ)
(36)
When assuming that the adsorption system is close to equilibrium (i.e., θ(e) ≈ θ) from eq 36, we obtain eq 37.
dθ λ-θ λ - θ(e) ) (e) dτ λ - θ λ-θ
(37)
k1 )
Klsc(e)(1 - θ(e))
(40)
c(in)V/Nm - θ(e)
Applying the approximation λ ≈ θ(e) means that N (e) f c(in)V (i.e., that almost all of the sorbate is removed from the solution in the course of experiment). Thus, on the ground of the SRT fundamental approach, good applicability of the Lagergren equation at longer adsorption times should be observed even in the case of the “solid dominated” systems and not only in the case of the volume dominated ones, as it was suggested by our analysis of the origin of the Lagergren equation, based on the classical absolute rate theory approach. Our model investigation shows that there are no significant differences between the more exact kinetic isotherm (eq 38) and the Lagergren eq 39 for both solid and volume dominated systems. 5. The Principles of SRT for the Kinetics of Solute Adsorption at Strongly Heterogeneous Solid Surfaces. One essential assumption inherent in the SRT is that of a quasiequilibrium on the solid surface. It is assumed that all correlation functions in the adsorbed phase are the same functions of the surface coverage θt as they would be at full equilibrium and the same surface coverage. At equilibrium we have eq 41,
µs ) µb ) µob + kT ln c(e)(θ(e) t )
(41)
and c(e)(θ(e) t ) can be calculated from the equilibrium isotherm equation. Thus, according to SRT, the kinetics of adsorption on heterogeneous surfaces should be expressed by the following (eq 42),
[ (
)
µb - µs(θt) dθt (e) ) Klsc(e)(1 - θ(e) t (c )) exp dt kT µs(θt) - µb exp kT
(
)]
(42)
where µb is expressed by eq 22, and µs can be calculated using eq 41 and a proper equation of the equilibrium adsorption isotherm. The equilibrium adsorption isotherm (eq 29) can be rewritten as eq 43.
(
)
θ(e) 1 t (m - l) - m c ) exp K kT (e)
(43)
The above differential equation can be easily solved with the boundary condition θ(τ ) 0) ) 0 to yield eq 38.
From eqs 41 and 43 we have eq 44.
θ(τ) )
µb(c) - µs(θt) ) (kT ln c) + (kT ln K) + m - θt(m - l) (44)
λ-
(
x
(λ - θ(e))2 + (2λ - θ(e))θ(e) exp -
2τ λ - θ(e)
)
(38) Then, the rate of adsorption reads as shown in eq 45.
A simple theoretical analysis shows that when the value of the λ parameter is close to θ(e), then this equation becomes close to the following Lagergren expression (eq 39).
[
(
)
(39)
- θt(m - l) + m dθt ) Klsc(e)(1 - θ(e) t ) Kc exp dt kT θt(m - l) - m 1 exp (45) Kc kT
Then, the theoretical interpretation of the k1 coefficient in eq 5 is as follows (eq 40).
Using the transformed equilibrium isotherm (eq 29), we obtain eq 46.
ln(θ(e) - θ(τ)) ) ln θ(e) -
τ λ - θ(e)
(
)]
Kinetics of Solute Adsorption at Solid/Solution Interfaces
[ ( (
)
dθt (θ(e) c t - θt)(m - l) ) Klsc(e)(1 - θ(e) ) exp t (e) dt kT c (θ(e) t - θt)(m - l) c(e) exp c kT
)]
(
c c(e)
)
c
(in)
Nmθ(e) t V
(46)
t
m - l kT
(
1+
x
≈ exp
λ - θ(e) t
λ-
)
-1
x θ(e) t
(48)
(49)
[(
dθt 1 ) exp x γ + dτ λ - θ(e) t
)]
(55)
≈ exp -
x
(56)
λ - θ(e) t
-
[ (
1 λ - θ(e) t
)] [( )] ( [ ( ( ) (
exp - x γ +
1
1 1 γ+ 3 λ - θ(e)
(52)
There5 we have shown that the Lagergren equation can be derived from eq 51. However, in the real kinetic experiments, noticeable changes in the bulk solution are commonly observed in the course of experiment, so c should be treated, in general, as a function of θt(t), as defined in eq 33. We show here that the Lagergren equation is also derived for that more general case of systems that are not necessarily volume dominated. It appears that the only essential condition is that the system must not be far from equilibrium, just like in the case of a homogeneous solid surface considered in the previous section. When the relation c/c(e) ) 1 is no longer fulfilled, one has to use the kinetic equation (eq 50), which takes into account the changes of solute concentration over time. This equation can then be written in the following form (eq 53),
)
)
We can see that the second term of the expansion vanishes, and the terms of order [x3] and higher may be much smaller and, therefore, neglected. Then we arrive at the Lagergren equation in its differential form (eq 59),
(
(51)
The above differential equation can be solved analytically. When the boundary condition θt(τ ) 0) ) 0 is assumed, eq 52 is the result.
θ(e) t
)2 γ+
1
x+ λ - θ(e) t 3 5 1 1 x3 + γ+ x5 + ... (58) (e) 60 λ - θt
λ-
t
In one of our previous papers5 we assumed that the adsorption system is volume dominated; therefore, we had c/c(e) ) 1. Then, eq 46 takes the form shown in eq 51.
(57)
-
t
λexp[ - γ(θ(e) t - θt)] (50) λ - θt
)]
After expanding the exponents into the Taylor series around x ) 0, we have eq 58.
dθt 1 ) exp x γ + dτ λ - θ(e)
θ(e) t
θt(τ) )
x
λ - θ(e) t
exp - x γ +
t
1 - ln[coth(γτ + arccoth eγθt(e))] γ
( ) ( )
From eqs 53, 55, and 56 we obtain eq 57.
λ - θt dθt exp[γ(θ(e) ) t - θt)] dτ λ - θ(e)
θ(e) t
(54)
The following approximations can be applied when the system is not far from equilibrium, that is, when x f 0 (eqs 55 and 56).
(47)
we can write eq 46 in the following form.
dθt (e) ) exp[γ(θ(e) t - θt)] - exp[ - γ(θt - θt)] dτ
e-γx (53)
x ) θ(e) t - θt
and using the notation in eq 49,
γ)
t
)
-1
where x is defined by eq 54.
1+
Introducing the dimensionless time τ (eq 48),
τ ) K′ls t ) Klsc(e)(1 - θ(e) t )t
) (
dθt x x eγx - 1 + ) 1+ (e) dτ λ-θ λ - θ(e)
As mentioned in the previous section, when noticeable changes in the bulk solute concentration are observed in a kinetic experiment, c in eq 46 is to be treated as a function of the adsorbed amount (eq 33). The ratio c/c(e) is then given by equation 47.
N m θt c(in) V
J. Phys. Chem. C, Vol. 111, No. 41, 2007 15105
)
dθt 1 )2 γ+ (θ(e) t - θt) (e) dτ λ-θ t
(59)
with the following interpretation of the k1 coefficient.
(
k1 ) 2
)
m - l 1 + (in) Klsc(e)(1 - θ(e)) (60) kT c V/Nm - θ(e) t
Thus, it is to be emphasized that, when using SRT approach, it is possible to derive the Lagergren equation for both homogeneous and heterogeneous models of a solid surface. However, it should also be emphasized that both derivations are proper only for the nonequilibrium surface coverages that are close to their equilibrium value θ(e) or θ(e) t . This is because neglecting the higher terms in our Taylor expansion (eq 58), which leads to the Lagergren equation, is justified only when the system is close to equilibrium, that is, c(in) ≈ c(e) or θ ≈ θ(e) in eq 36. Analysis of Experimental Data and Conclusions Before we start presenting our theoretical analysis of some available experimental data, let us briefly summarize the obtained theoretical results.
15106 J. Phys. Chem. C, Vol. 111, No. 41, 2007 The square-root dependence of the adsorbed amount on time or the Lagergren equation have been commonly applied to represent the adsorption kinetics at short and long adsorption times, respectively. Meanwhile, as we have shown in our two previous publications,6,9 one will face a similar situation in the systems in which the adsorption rate is controlled by the rate of the adsorption reactions occurring at a solid/solution interface. This shows how confusing the conclusions found in many papers and drawn from the applicability of various simple kinetic equations may be. At the same time, drawing a correct conclusion about the model of adsorption kinetics is very essential for proper design and operation of an adsorption installation. Carrying out experiments with adsorbent particles having different sizes (radius, r) may also be a way to distinguish between the diffusional and the surface reaction kinetic models. Provided that particles of different sizes have identical surface properties, the independence of the observed kinetics on particle size should speak for surface reaction kinetic models. This, however, should be confirmed by simultaneous measurements of adsorption equilibria. Then, it is to be emphasized, a reverse behavior of the observed kinetics may not necessarily speak for the diffusional model to be accepted. This is because particles of different sizes may exhibit somewhat different adsorption properties.19-24 Moreover, probably for practical reasons, most of the kinetic experiments are carried out with particles of one kind, and only such data are usually available for analysis. So, when trying to find a way of drawing reliable conclusions about the observed kinetics, we have focused our attention on the region of longer adsorption times, at which an adsorption system is not far from equilibrium. Then, the generally very complicated kinetic expressions reduce to some simple ones, which can be easily applied for the theoretical interpretation of the observed experimental data. Next, when taking the longer times into consideration, we have decided to predict which would be the behavior of adsorption kinetics when we accept the most popular Langmuir model of adsorption (monolayer, ideal, one-site-occupancy adsorption). Our theoretical studies have been based on applying both the classical TAAD approach to adsorption kinetics and the new fundamental SRT approach to the interfacial kinetics. Both approaches suggest that, for the Langmuir model of adsorption, the popular Lagergren equation should be the limiting form of all kinetic equations in the systems approaching the thermodynamic equilibrium. The inapplicability of the Lagergren equation, or a better applicability of other kinetic equations, may be due either to serious departures from equilibrium conditions or to the inapplicability of Langmuir model of adsorption. However, although the Lagergren equation is also the limiting form for the diffusional model, we have shown that the theoretical interpretation of the determined kinetic coefficients is different for the diffusional and the surface reactions kinetic models, respectively. Also, we have shown that a simultaneous experimental study of equilibrium adsorption isotherm is essential for a proper interpretation of these kinetic parameters. Therefore, looking in literature for a proper set of experimental data for an illustrative theoretical analysis, we selected the data for which the following two conditions would be fulfilled:(1) at longer adsorption times the observed adsorption kinetics can be represented well by the Lagergren equation, and (2) the simultaneously measured equilibrium adsorption isotherms would correspond to a similar region of surface coverages, as those monitored in the kinetic experiment.
Rudzinski and Plazinski
Figure 1. The adsorption isotherm of MCB onto pine sawdust, measured by O ¨ zacar and Sengy´l.25 The measured data for the equilibrium isotherm (O) are correlated by using the commonly applied linear representation for the Langmuir isotherm eq 7. The solid lines are the theoretical values calculated by accepting the following values of parameters: Nm ) 270.76 mg/g and KL ) 0.006034 dm3/mg.
With such conditions in mind we again focused our attention on the MCB/pine sawdust adsorption system, already considered in our previous publication.6 Another paper considered by us, by M. O ¨ zacar and I. A. Sengy´l, reports on both kinetic and equilibrium isotherms of adsorption of metal complex dye (MCB) onto pine sawdust.25 The equilibrium adsorption isotherm measured in this system can be fairly well correlated by the Langmuir equation, as shown in Figure 1. Having determined the parameters of the equilibrium adsorption isotherm, we used them to fit the experimental kinetic plots N(t) versus t by adjusting only the values of the parameters Kls, when eq 36 was applied, or Dπ2/r2, when eq 1 was applied. The comparison of these two fits led us to an intriguing observation that both the diffusional and the surface reaction kinetic models gave almost equally good fits. In both cases, the fitting procedure required adjusting just one free parameter, and as for the error function, the standard deviation was accepted. It has long been known and discussed in literature that the choice of an error function and data representation may affect the determined function parameters and even some basic conclusions drawn from such best-fit exercises. So, following the idea of this paper, we have decided to use the linear Lagergren regression for the kinetic data points close to the equilibrium values N (e) (eq 61).
ln(N (e) - N(t)) ) β - κt
(61)
Having determined the N (e) values from the equilibrium isotherm, we plotted the experimental ln(N (e) - N(t)) values as the function of time. However, this procedure was used only for those N(t) data points that were close to the equilibrium value N (e) (to our personal feeling, such a situation is observed for the contact times higher than 20 min). These plots were then subjected to the linear regression (eq 61). The obtained results are shown in Figure 2, whereas Table 1 collects the values of the best-fit β and κ parameters determined in that way. By looking at Figure 2 one can see a fairly good linearity in the ln(N (e) - N(t)) versus t plots for all five initial solute concentrations. This indicates a good applicability of the Lagergren equation. The comparison of the values of the β parameter found in that linear regression with its two possible theoretical interpretations (eqs 15 and 16) seems to make the drawing of some conclusions possible as to whether surface reaction or intraparticle diffusion is the likely rate-determining step. Namely, the determined values of the β parameter, collected in Table 1 are closer to the ln N (e) + ln 6/π2 (i.e., to βD) defined in eq 15, rather than to ln N (e). This would speak for the applicability of the diffusional model rather than of the surface
Kinetics of Solute Adsorption at Solid/Solution Interfaces
J. Phys. Chem. C, Vol. 111, No. 41, 2007 15107 TABLE 2: The Values of Dπ2/r2 Parameters, Found in Our Fitting Exercises Presented in Figure 3 c(in) [mg/dm3]
Dπ2/r2 [1/min]
50 100 150 200 300
0.031 0.055 0.063 0.051 0.050
TABLE 3: The N0/N (e) Values Calculated from eq 70 for the N (e) and β Values Collected in Table 1
Figure 2. The Lagergren plots (eq 61) drawn for the experimental kinetic data reported by M. O ¨ zacar and I. A. Sengy´l25 on the MCB/ pine sawdust adsorption system. The kinetics of adsorption was studied at the five initial solute concentrations: c(in) ) 50 mg/dm3 (O), 100 mg/dm3 (]), 150 mg/dm3 (0), 200 mg/dm3 (4), and 300 mg/dm3 (*). The linear regression (eq 61) was made for the data points measured for the contact times higher than 20 min, and the N (e) values were determined from the equilibrium adsorption isotherm.
c(in) [mg/dm3]
N0/N (e)
50 100 150 200 300
0.75 0.78 0.82 0.82 0.80
Table 2, are not close to those obtained from the linear regression (eq 61, Table 1). Generally, they are three times higher. This would suggest that the value of Dπ2/r2 is not constant for all adsorption times but is higher for the initial fast kinetics. So, it seems reasonable to assume that we may have to deal with a two-step-kinetic process. A first fast initial kinetics is next followed by the slower kinetics, which corresponds to the linear Lagergren plots in Figure 2. The fact that the β values in Table 1 are closer to βD than to βL strongly suggests that the slower second kinetics is of a diffusional character. At the same time, there remains to be explained the intriguing question as to why the β values are much lower than the βD values. We propose the following hypothesis to explain that intriguing question. Namely, the second kinetic process sees the already adsorbed amount as a kind of preadsorption. Assuming that N0 in eqs 19 and 20 is the amount adsorbed during the first fast kinetic process, we arrive at the following generalization of eq 4,
ln(N (e) - N(t)) ) Figure 3. Kinetics of MCB adsorption on pine sawdust, studied at the five solute initial concentrations: c(in) ) 50 mg/dm3 (O), 100 mg/ dm3 (]), 150 mg/dm3 (0), 200 mg/dm3 (4), and 300 mg/dm3 (*). The theoretical kinetic isotherms (ss) were calculated from the full form of the diffusional equation (eq 1) using the best-fit values of the Dπ2/ r2 parameters, collected in Table 2.
TABLE 1: The Values of the Best-fit Parameters Found in the Linear Regression Equation,a Presented in Figure 2 c(in) βD ) ln N (e) κ N (e) [mg/g] [mg/dm3] [1/min] (experimental) βL ) ln N (e) + ln 6/π2 50 100 150 200 300 a
0.01075 0.01504 0.01028 0.01412 0.01994
29.3 56.3 82 103 135
3.378 4.031 4.407 4.635 4.905
2.880 3.533 3.909 4.137 4.408
β 2.003 2.520 2.685 2.944 3.292
Equation 61.
reaction model. Thus, the κ parameter appearing in eq 61 should be identified with Dπ2/r2. This fact would suggest correlating the whole measured experimental kinetic isotherm by applying the full form of the diffusional equation (eq 1). We did it using the N (e) values determined from the equilibrium adsorption isotherm, and then we adjusted the values of the Dπ2/r2 parameters to get the best fit. The results of applying this procedure are presented in Figure 3 and in Table 2. One can see quite good fits in Figure 3, but the values of the best-fit parameters Dπ2/r2, determined now and collected in
ln N (e) + ln
(
)
2 π 2 π N0 6 Dπ 2 + ln - 2 t (62) 2 (e) 6 π r 6N
for the second diffusional kinetics taking place under the conditions which are not far from the equilibrium conditions. So, the β values collected in Table 1 now have the following interpretation (eq 63).
β ) ln N (e) + ln
(
2 6 π 2 π N0 + ln 6 π2 6N (e)
)
(63)
Having known N (e) and β values, we may calculate the N0 values corresponding to all of the initial concentrations. Then, it is very interesting to see that the N0/N (e) values are fairly the same for all of the initial solute concentrations. They are collected in Table 3. Table 3 would suggest that the second kinetic process starts to dominate when about 80% of the equilibrium amount is adsorbed. One can easily see that this is the region in which the data points taken for correlation by the Lagergren equation lie. Concluding, however, that the data points lying below this region correspond to another kind of kinetics does not mean that one can try another Lagergren plot to correlate these data. These data are far from equilibrium, so they cannot be correlated by the Lagergren equation. Our numerical exercises, presented in our previous paper,6 show that the assumption of both
15108 J. Phys. Chem. C, Vol. 111, No. 41, 2007
Figure 4. The Lagergren plots drawn for the experimental kinetic data published by C. W. Cheung et al.26 in their paper on adsorption of the cadmium ions onto bone char. The adsorption kinetics were studied at the five different solute initial concentrations: c(in) ) 2.15 mmol/dm3 (0), 2.69 mmol/dm3 (]), 3.17 mmol/dm3 (4), 4.18 mmol/dm3 (X), and 5.24 mmol/dm3 (*). The linear regression (eq 61, ss) was made for the data points measured at contact times higher than 3 h, and the obtained values of the parameters are collected in Table 4.
diffusional and surface reaction kinetics leads to a comparable fit of these data. One essential difficulty is that we have at our disposal only a few data points recorded for this initial fast kinetics. So, drawing some more deeply going conclusions seems to be difficult, in view of the amount of the experimental information reported on this system by O ¨ zacar and Sengy´l.25 When looking for an adsorption system with a typically heterogeneous surface, we have focused our attention on the paper by C. W. Cheung et al., reporting on sorption of cadmium ions by bone char.26 The authors report that the measured equilibrium adsorption isotherm can be correlated well by the Langmuir-Freundlich equation, which would suggest a system in which the solid surface is energetically heterogeneous. However, the procedure of determining the equilibrium data applied by Cheung et al. led them to N (e) values somewhat different from those found by these authors testing the applicability of various kinetic equations to fit their experimental kinetic isotherms. Therefore, we have decided to treat N (e) in eq 61 as the parameter leading to the best linear regression of the kinetic data close to the equilibrium values. After digitizing the experimental data presented in Figure 2 in the original paper by Cheung et al., we used eq 61 to arrive at the best linear regression of the values of ln(N (e) - N(t)) plotted as the linear function of time but only for those data points that were measured for the contact times higher than 3 h (when the system was close to equilibrium). The results of these fits are shown in Figure 4, and the values of the obtained best-fit parameters are collected in Table 4. By looking at Table 4 one can see that the determined values of the abscissas are very close to the value of βD and not βL. This strongly suggests a good applicability of the diffusional kinetic model for this system. Accordingly, it suggests that the full diffusional equation (eq 1) could be used for the correlation of the whole kinetic isotherm. Our numerical exercises have confirmed that expectation. Taking the N (e) values (Table 4), determined from the Lagergren
Rudzinski and Plazinski
Figure 5. Kinetics of adsorption of the cadmium ions onto bone char, studied by Cheung et al.44 at the five different solute initial concentrations: c(in) ) 2.15 mmol/dm3 (0), 2.69 mmol/dm3 (]), 3.17 mmol/ dm3 (4), 4.18 mmol/dm3 (X), and 5.24 mmol/dm3 (*). The theoretical kinetic isotherms (ss) plotted in panel A were calculated from eq 1 using the N (e) values, leading to the best linear Lagergren regressions shown in Figure 4, whereas those plotted in panel B were also calculated from eq 1, but the accepted values of N (e) were those determined by Cheung et al. in their procedure of arriving at the equilibrium adsorption isotherm. In both cases, the Dπ2/r2 parameters were treated as best-fit parameters, and their values are collected in Table 5.
TABLE 4: The Values of the Parameters Found in the Correlation of the Measured Kinetic Data, Shown in Figure 4a βD ) ln N (e) N (e) [mmol/g] βL ) ln N (e) + ln 6/π2
c(in) [mmol/dm3]
κ [1/h]
2.15 2.69 3.17 4.18 5.24
0.2444 0.2474 0.2391 0.2518 0.2711
-1.033 -0.9188 -0.8233 -0.7340 -0.6714
0.356 0.399 0.439 0.480 0.511
β
-1.5305 -1.4165 -1.3210 -1.2317 -1.1691
-1.4500 -1.2378 -1.1876 -1.1301 -1.1220
a First, the values of N (e) were adjusted to arrive at the best linear regression, and then the values of κ and β were found from that regression.
TABLE 5: The Values of the Parameters Used in the Fits Presented in Figure 5 panel A c(in)
2/r2
panel B (e)
2/r2
[mmol/dm3]
Dπ [1/h]
N [mmol/g]
Dπ [1/h]
N (e) [mmol/g]
2.15 2.69 3.17 4.19 5.24
0.231 0.220 0.220 0.235 0.265
0.356 0.399 0.439 0.480 0.511
0.194 0.182 0.191 0.191 0.205
0.378 0.425 0.460 0.515 0.556
plots and properly adjusting the Dπ2/r2 values, we arrived at the excellent agreement shown in Figure 5A. The adjusted values of the Dπ2/r2 parameter are collected in Table 5. These values are not far from those determined from the Lagergren plots (the values in Table 4). Our numerical exercises also showed that the fit of the whole kinetic isotherm by the full diffusional equation (eq 1) is very sensitive to the choice of the N (e) values. Figure 5B shows that using the N (e) values determined by Cheung et al. in their procedure of arriving at equilibrium conditions does not lead to as good a fit of the kinetic isotherms as using the values of N (e) determined from their kinetic isotherms. So, it is interesting
Kinetics of Solute Adsorption at Solid/Solution Interfaces
Figure 6. The linear correlations by the Temkin equation (eq 29) of the N (e) values found from the Lagergren plots presented in Figure 4 (0) and the N (e) values measured by Cheung et al. (9) and presented in Table 2 in their original paper.26 The coefficients of these linear regressions are as follows: for (0): NmkT/(m - l) ln K + Nmm/(m - l) ) 0.4340 mmol/g and NmkT/(m - l) ) 0.07922 mmol/g; and for (9): NmkT/(m - l) ln K + Nmm/(m - l) ) 0.47651 mmol/g and NmkT/(m - l) ) 0.08389 mmol/g.
to note that applying other kinetic equations also led Cheung et al. to generally lower values of N (e) than those determined in their procedure of interpolation into the equilibrium conditions. One may consider the following explanation for that. Namely, the N (e) values reported by Cheung et al. as the equilibrium ones were the N(t) values measured after a very long time, much longer than the period of 10 h within which the kinetics was monitored. So, the slightly higher equilibrium values determined in such a way may have their source in slow kinetics of another kind accompanying the main kinetic process. This might, for instance, be diffusion into the interior of very narrow pores. Although the contribution of such slow kinetics may be negligible during the first 10 h, it will still contribute to N(t) even when the main kinetic process terminates. However, this is always the main kinetic process that is essential for industrial applications of adsorption, in which sorbent and solution are brought into contact only for a limited period of time. Therefore, our theoretical considerations in this paper have been based on a model of one kinetic process governing the rate of adsorption under conditions that are not far from equilibrium. It appears, however, that the studies of the adsorption kinetics in this region may provide an essential information about the whole kinetic process. There is still another proof speaking for more proper determination of the N (e) values from the kinetic isotherms measured by Cheung et al. Figure 6 shows that the N (e) values determined from the kinetic Lagergren plots can be very well correlated by the Temkin theoretical isotherm (eq 29). This also speaks for the correctness of the accepted model of a system with an energetically heterogeneous solid surface. Summary In trying to find a way of drawing conclusions about the kinetic model governing the rate of adsorption, we have focused our attention on the behavior of adsorption systems when they are close to equilibrium. Their theoretical analysis shows that for both the diffusional and the surface reaction models the kinetics of adsorption should be represented by the Lagergren equation. In the case of the surface reaction model, our theoretical analysis concerned the most popular Langmuir model of adsorption. That means an ideal, monolayer, one-siteoccupancy of adsorption. An assumption that it may be an energetically heterogeneous lattice of adsorption sites does not change the basic conclusion that the kinetics of adsorption in all of the systems that are not far from equilibrium is still
J. Phys. Chem. C, Vol. 111, No. 41, 2007 15109 described by the Lagergren equation. However, as in the case of the diffusional model, the Lagergren equation is only the limiting form of more general kinetic equations, corresponding to the surface reaction kinetic model. Therefore, it cannot be applied to correlating the kinetic data for all adsorption times. The theoretical derivation of the Lagergren equation presented here shows that the above basic conclusion is also true in the systems where noticeable changes of the solute concentration are observed in the course of a kinetic experiment. Studying kinetic data that are not far from the equilibrium values creates an interesting chance of distinguishing between the diffusional and the surface reaction models. Although the Lagergren equation is to be used in both cases, the theoretical interpretation of the coefficients in that equation appears to be different for the two different kinetic models. A comparison of the theoretically predicted values of these coefficients with the experimentally determined ones seems to offer a chance to a draw conclusion of which kinetic model should be assumed. We have demonstrated this possibility by analyzing the behavior of the two adsorption systems (i.e., MCB/pine sawdust25 and Cd2+/ bone char26). The method based on analysis of the coefficients in the Lagergren equation is, of course, applicable to the adsorption systems that are not far from equilibrium. We do not exclude the complicated, but possible, case when the model of adsorption changes in the course of experiment. However, our theoretical analysis shows that the inapplicability of a single Lagergren equation to describe the whole adsorption process cannot be treated as proof for the changing model of adsorption kinetics. Our theoretical analysis shows that, even in the case when the kinetic model does not change, the whole kinetic isotherm is to be described by the equations that are much more complicated than the commonly used lumped equations. At the same time, using these simple equations may lead to interesting conclusions, but the theoretical origin of these simple equations and the range of their applicability is to be properly recognized. Here, we have focused on the general applicability of the Lagergren equation to represent the kinetics of adsorption in the systems that are not far from equilibrium. It appears, however, that studies of the kinetics in this region may provide essential information about the nature of the whole kinetic process. References and Notes (1) Lagergren, S. Kungliga SVenska Vetenskapsakademiens. Handlingar 1898, 24 (4), 1. (2) Ho, Y. S.; McKay, G., Wat. Res. 2000, 34 (3), 735. (3) Ho, Y. S.; McKay, G. Process Biochem. 1999, 34, 451. (4) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (5) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514. (6) Rudzinski, W.; Plazinski, W. Appl. Surf. Sci. 2007, 253, 5827. (7) Ho, Y. S. Wat. Res. 2006, 40, 119. (8) Suzuki, M. Adsorption Engineering; Kodansha: Tokyo, 1990. (9) Rudzinski, W.; Panczyk, T.; Plazinski, W. J. Phys. Chem. B 2005, 109, 21868. (10) Mohanty, K; Das, D.; Biswas, M. N. Adsorption 2006, 12, 119. (11) Chandrasekhar, S.; Pramada, P. N. Adsorption 2006, 12, 27. (12) Dogan, M.; Alkan, M. Chemosphere 2003, 50, 517. (13) Ho, Y. S.; McKay, G. Wat. Res. 1999, 33 (2), 578. (14) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (15) Rudzinski W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces, Academic Press: London, 1992. (16) 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. (17) Rudzinski, W.; Panczyk, T. J. Non-Equilib. Thermodyn. 2002, 27, 149. (18) Ward, C. A.; Findlay, R.D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599.
15110 J. Phys. Chem. C, Vol. 111, No. 41, 2007 (19) Zhou, Y.; Liu, R.; Tang, H. J. Colloid Interface Sci. 2004, 270, 37. (20) Wang, S; Jin, X., Bu, Q.; Zhou, X.; Wu, F. J. Hazard. Mat. 2006, A128, 95. (21) Guibal, E.; Larkin, A.; Vincent, T.; Tobin, J. M. Ind. Eng. Chem. Res. 1999, 38, 4011. (22) Ho, Y. S.; Huang, C. T.; Huang, H. W. Process Biochem. 2002, 37, 1421.
Rudzinski and Plazinski (23) Feng, D.; Aldrich, C. Hydrometallurgy 2004, 73, 1. (24) Chiuo, M.-S.; Li, H.-Y. J. Hazard. Mat. 2002, B93, 233. (25) O ¨ zacar, M.; Sengy´l, I. A. Bioch. Eng. J. 2004, 21, 39. (26) Cheung, C. W.; Porter, J. F.; McKay, G. Wat. Res. 2001, 35 (3), 605.
