Aqueous Interfaces: Searching

Langmuir 2008, 24, 5393-5399

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Kinetics of Solute Adsorption at Solid/Aqueous Interfaces: Searching for the Theoretical Background of the Modified Pseudo-First-Order Kinetic Equation Władysław 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, Uniwersytet Marii Curie-Skłodowskiej, pl. M. Curie-Skłodowskiej 3, 20-031 Lublin, Poland ReceiVed January 7, 2008. ReVised Manuscript ReceiVed February 11, 2008 It is shown that the modified pseudo-first-order (MPFO) kinetic equation proposed recently by Yang and Al-Duri simulates well the behavior of the kinetics governed by the rate of surface reaction and described by our general kinetic equation, based on the statistical rate theory. The linear representation with respect to time, offered by the MPFO equation seems to be a convenient tool for distinguishing between the surface reaction and the diffusional kinetics. Also, a method of distinguishing between the surface reaction and the intraparticle diffusion model based on analyzing the initial kinetic isotherms of sorption is proposed. The applicability of these procedures is demonstrated by the analysis of adsorption kinetics of the reactive yellow dye onto an activated carbon.

Introduction Adsorption at solid/solution interfaces is of crucial importance for life on our planet, a large variety of industrial processes, and a variety of processes in environmental protection. In the case of the environmental processes, adsorption is an efficient and economically feasible process for the treatment of wastewater containing chemically stable pollutants such as dyes or poisoning ions, for instance. Here the solution and the sorbent are brought into contact for a limited period of time, so not only the equilibrium features of such solid/solution systems but also their evolution in time are of fundamental importance for the sorption technique to be an effective process for wastewater treatment. So it is no surprise that so many papers were published on the kinetics of sorption at solid/solution interfaces. It is very impressive to note that the first ever theoretical paper on adsorption was that proposing a theoretical expression to describe the kinetics of adsorption at the solid/solution interface. This was the paper by Lagergren, published in 1898 in which the kinetic equation, commonly called the pseudo-first-order equation was proposed.1 However, even more impressive is the fact that, after more than one century, this equation is still one of the most popular and widely used equations to correlate the experimentally monitored kinetic isotherms of adsorption. At the same time, this kinetic equation has remained an essentially empirical one until very recently. It was Azizian2 who proposed its first derivation based on the classical fundamental TAAD approach (theory of activated adsorption/desorption), whereas Rudzinski and Plazinski3 derived it for the first time by applying the new fundamental SRT approach to the kinetics of interfacial transport (statistical rate theory). The Lagergren pseudo-first-order equation leads to the conclusions that the logarithm of the difference between the equilibrium and the monitored adsorbed amount should be a linear function * Corresponding author, [email protected]. † Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences. ‡ Department of Theoretical Chemistry, Faculty of Chemistry, Uniwersytet Marii Curie-Skłodowskiej. (1) Lagergren, S. Kungliga SVenska Vetenskapsakademiens. Handlingar 1898, 24, 1. (2) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (3) Rudzinski, W.; Plazinski, W J. Phys. Chem. B 2006, 110, 16514.

of time. Such a simple linear representation was a convenient and therefore very attractive tool to correlate the experimental kinetic isotherms. Common use of the Lagergren equation also led to discovery of various discrepancies between predicted and observed behaviors of the kinetic adsorption isotherms. Typical observations were follows: The Lagergren kinetic equation was not able to correlate the whole kinetic isotherm, i.e., the kinetic data monitored at both short and long adsorption times. The linear Lagergren plot could commonly correlate well adsorption data monitored at longer adsorption times when a system was not far from equilibrium but failed to describe the initial region corresponding to short adsorption times. The following three hypotheses have been considered to explain these typical deviations: (1) Monitored kinetics is a multistep process in which switching takes place from one kinetic mechanism to another. (2) The monitored kinetic isotherm corresponds to various processes occurring simultaneously and one of them dominates in a certain region of times. (3) The applied Lagergren equation is a “lumped” form of a more adequate but also sophisticated equation which eventually might correlate kinetic data better. The scientists accepting the first of the above-mentioned hypotheses applied different Lagergren plots to correlate different parts of the monitored kinetic isotherm. While following the second hypothesis, the scientists considered the whole kinetic process as composed of kinetic processes occurring on different parts of a solid surface. In response to the third hypothesis, a variety of other kinetic equations was proposed to correlate experimental data. Some of them were purely empirical and intuitively related to the kinetic mechanism when the adsorption kinetics is governed by the rate of surface reaction. These have been the Elovich and the pseudo-second-order kinetic isotherms, for instance. Also, there have been proposed equations corresponding to the kinetic mechanism when the adsorption kinetics is governed by the rate of intraparticle diffusion. The square-root dependence of the adsorbed amount on time generally has been

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believed to be a solid check that the adsorption kinetics is controlled by the intraparticle diffusion. Meanwhile, recently it has been shown by Rudzinski and Plazinski that such a square-root dependence in the initial region of adsorption times may also be typical of kinetics controlled by the rate of surface reaction. They arrived at this conclusion by accepting the popular Langmuir model of adsorption and applying the new fundamental SRT approach to adsorption kinetics.4 While applying that new SRT approach, Rudzinski and Plazinski5 have also shown that the Lagergren equation is the limiting form of the kinetic isotherm corresponding to the Langmuir model of adsorption, when the system is not far from equilibrium. Therefore, we may summarize the present situation as follows. Using the new fundamental SRT approach to adsorption kinetics, we have arrived at much more adequate and accurate kinetic equations to represent the kinetics which is governed by the rate of surface reactions. That more adequate but also more complicated equation reduces to the square-root dependence on time for short adsorption times and to the Lagergren equation for long times. These simple equations are very attractive for being used to correlate experimental data, but their use is limited to short and long adsorption times, respectively. Here we are going to show that the new modified pseudo-first-order kinetic equation proposed recently on an empirical basis by Yang and Al-Duri6 reproduces very well the exact solution of our kinetic equation in the whole region of adsorption times. It means that this empirical equation reproduces well the behavior of our theoretical kinetic isotherm in the whole region: from short initial times to long times when the system is close to equilibrium. At the same time the simplicity of that empirical equation makes it a very attractive and convenient tool to correlate experimental data. The theoretical background of this modified pseudo-first-order equation will be discussed, and its applicability will be presented by an appropriate analysis of some experimental data reported in the literature.

in the bulk phase (b) and the phase adsorbed on the surface (a). Further, k and T denote the Boltzmann constant and absolute temperature, as usual. For the bulk phase the following expression is accepted

µb ) µ°b + kT ln c

where c is the concentration of the solute in the bulk phase. The expression for µa is derived from the equilibrium adsorption isotherm. This is because one fundamental assumption of the SRT approach is that the adsorbed phase is in a quasi-equilibrium. It means that all the correlation functions are the same as they would be at equilibrium at the same surface coverage. So let us consider the popular Langmuir model of adsorption and its corresponding adsorption isotherm:

θe )

dθ ) k1(θe - θ) dt

(1)

where θ(t) is the monitored fractional surface coverage, θe is that surface coverage at equilibrium, i.e.

θe ) lim θ(t)

[

dθ 1-θ 1 θ ) Klsce(1 - θe) KLc dt θ KLc 1 - θ

[ (

)

(

µb - µa µa - µb dθ - exp ) Kls′ exp dt kT kT

)]

{

(5)

(6)

and accepting relation 4 we can transform eq 5 to the following form

dθ θe(λ - θ)(1 - θ) θ(λ - θe)(1 - θe) ) dτ θ(λ - θe)(1 - θe) θe(λ - θ)(1 - θ)

(7)

Now, let us write for comparison the rate equation offered by the classical TAAD approach

dθ ) Kac(1 - θ) - Kdθ dt

(8)

in which Ka and Kd are the temperature-dependent constants such that KL ) Ka/Kd Further, let us consider the limiting features of these equations when t f ∞ and t f 0, respectively. For the case t f ∞, we expand the derivatives dθ/dt (or dθ/dτ) into their Taylor series around θ ) θe. Then from the TAAD equation (8) we obtain

θfθe

(2)

(4) Rudzinski, W.; Panczyk, T.; Plazinski, W. J. Phys. Chem. B 2005, 109, 21868. (5) Rudzinski, W.; Plazinski, W. J. Phys. Chem. C 2007, 111, 15100. (6) Yang, X.; Al-Duri, B. J. Colloid Interface Sci. 2005, 287, 25. (7) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599.

]

λ ) cinV ⁄ Nm τ ) K ′lst ) Klsce(1 - θe)t

lim

where Kls′ is the rate of adsorption/desorption at equilibrium, whereas µb and µa are the chemical potentials of adsorbate (solute)

(4)

where K′ls ) Klsce(1 - θe). One can also take into account the set of technical parameters which characterize the conditions under which the kinetic experiment is carried out, i.e., the volume of solution (V), the solute initial concentration (cin), and the monolayer capacity (Nm) which is connected with the applied mass of adsorbent. Using the notation

tf∞

whereas k1 is a constant. The physical meaning of that constant remained unexplained for more than one century because the Lagergren equation has remained essentially an empirical one until very recently. Its first theoretical background was given by Azizian2 in 2004 and was based on the classical TAAD approach to adsorption kinetics. Two years later Rudzinski and Plazinski proposed its theoretical derivation based on the new fundamental SRT approach.3 Following that approach7

KLce 1 + KLce

In eq 4 KL is the temperature-dependent Langmuir constant whereas the subscript e refers to the equilibrium conditions. Assuming also that the bulk/surface exchange rate is proportional to the frequency of the collisions of the solute molecules with the available fraction of surface, one arrives at the following rate equation

Theory The famous Lagergren kinetic equation is usually written as follows1

(3)

dθ ) (Kace + Kd)(θe - θ) dt

(9)

whereas the SRT equation 7 leads to the following expression:

2(λ - θe2) dθ ) (θ - θ) + O[θe - θ]2 (10) θfθe dτ θe(λ - θe)(1 - θe) e lim

Integration of both eqs 9 and 10 with the boundary condition θ(t ) 0) ) 0 yields the following integral form of the Lagergren equation

The Modified Pseudo-First-Order Kinetic Equation

ln(θe - θ) ) ln θe - k1t

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The constant k1 is defined in the case of eq 9 as

k1 ) Kace + Kd

(12)

whereas in the case of eq 10:

k1 )

2Klsce(λ - θe2) θe(λ - θe)

(13)

Now let us note that also the popular intraparticle diffusion model leads to the Lagergren equation in the limit θ f θe Let us consider for that purpose the most widely used mathematical expression corresponding to this model8

(

∞

)

θ 1 6 Dn2π2 )1- 2 exp t θe π n)1 n2 r2

∑

(14)

Dπ2 6 t + ln θe + ln 2 r2 π

(15)

which suggests that adjusting properly the value of θe should make the experimentally determined ln(θe - θ) data a linear function of time. The tangent of such a linear plot will be equal to -Dπ2/r2 and its abscissa to (ln θe + ln (6/π2)). For the purpose of our further consideration, it is essential to emphasize that the value of θe parameter determined from the abscissa should be equal to that making the plot ln(θe - θ) a linear function of time. The important difference lies in the theoretical interpretation of the abscissas in eq 15 (βD)

βD ) ln θe + ln

6 π2

(16)

and in eq 11 (βL)

βL ) ln θe

(17)

It has been frequently emphasized in literature that for short times, when the fractional uptake θ f θe is less than around 0.3, eq 14 reduces practically to the following one

θ)

6θe r


Dtπ

(18)

As we have already stated, the Lagergren equation is the limiting form of both diffusional and the surface reaction kinetic equations at long adsorption times when the system approaches equilibrium. In the case of the surface reaction kinetic model, this conclusion was drawn by applying both the classical TAAD approach and the new fundamental SRT approach to adsorption kinetics. A somewhat different situation is faced when these two approaches are applied to predict the limiting form of the kinetic equations for short adsorption times, i.e., when t f 0. While applying the classical TAAD approach to adsorption kinetics, from eq 8, one arrives at the following limiting form of the surface reaction kinetic equation:

lim tf0

dθ ) Kac dt

(19)

Its integration with the boundary condition θ(t ) 0) ) 0 yields (8) Suzuki, M. Adsorption Engineering; Kodansha: Tokyo, 1990.

(20)

It means that the plot θ vs t should, at t f 0, become linear with tangent being a linear function of the initial solute concentration. One arrives at a dramatically different conclusion when the new SRT approach is applied to represent the adsorption kinetics governed by the rate of surface reactions.4 Let us remark that then, in the limit t f 0, both the diffusional kinetic model and the surface reaction kinetic model predict the same square-root dependence on time of the surface coverage. In the case of the surface reaction kinetic model, from eqs 4 and 5 we obtain:

lim tf0

where D is the intraparticle diffusivity and r is the particle radius. At long times, when θ f θe, eq 14 reduces to the Lagergren-like expression

ln(θe - θ) ) -

θ(t) ) Kact ) Kacint

(11)

dθ 1 ) Klsce(1 - θe)KLcin dt θ

(21)

Its integration with the boundary condition θ(t ) 0) ) 0 yields

θ ) √2Klsce(1 - θe)KLcint ) √2Klscinθet

(22)

Thus, at t f 0, tangent of this linear plot is always a linear function of the expression (cinθe)1/2. It also means that the square root dependence on time of the monitored surface coverage θ(t1/2) cannot be treated as a proof that the kinetics of sorption is governed by the rate of intraparticle diffusion. One should, however, remember that eq 21 does not apply at extremely low times and surface concentrations when the predicted rate of adsorption is higher than the flux of solute molecules to the surface. For such extreme low times and surface concentrations a more complicated procedure is to be applied.9 Nevertheless, drawing such double-logarithmic plots may be a source of interesting information concerning the first kinetic process. Provided that it could be governed by the rate of intraparticle diffusion, as follows from eq 18, the tangent should be a linearly increasing function of θe. Thus, while considering the first region of small adsorption times and surface coverages, we face a similar situation to that when the system is close to equilibrium. Namely, both the diffusional and the surface reaction kinetics are represented by the same kind of theoretical expression, and the difference lies only in the interpretation of their kinetic coefficients. In the case of the diffusional kinetics the tangent of the linear plot θ vs t1/2 will be a linear function of the equilibrium surface coverage θe, whereas in the case of the surface reaction model, it will be a more complicated function suggested by eq 22. Therefore, we may summarize the results of our considerations as follows: In the initial region of short adsorption times the square-root dependence of the adsorbed amount on time is the “lumped” expression valid for both the surface reaction and the diffusional kinetic models. For both kinetic models, the Lagergren expression is the “lumped” form of appropriate, more adequate, and sophisticated expressions, describing the rate of adsorption when the system approaches equilibrium. Distinguishing between these two kinetic models needs a careful analysis of the behavior of the coefficients in these lumped expressions. The square-root dependence on time or the Lagergren plots have been frequently used to correlate experimental kinetic isotherms. However, as the theoretical background of these lumped relations has remained unclear until recently, both their successful applications and the observed deviations could not receive solid interpretation. It seems that the main source of their popularity and common use were the simple correlations of (9) Panczyk, T. Phys. Chem. Chem. Phys. 2006, 8, 3782.

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experimental data offered by these lumped expressions. They made their use very attractive for many researchers. So no surprise that various attempts were made to modify these expressions to increase the range of their applicability. Here we would like to draw the readers’ attention to some very reasonable modification proposed by Yang and Al-Duri6 on an empirical basis. Very recently these authors proposed the following modification of the Lagergren equation

θe dθ ) KM (θe - θ) dt θ

(23)

where KM is a constant. We will call it the MPFO (modified pseudo-first-order) equation. Its integration with the boundary condition θ(t ) 0) ) 0 yields

ln(θe - θ) +

θ ) ln θe - KMt θe

(24)

Let us remark that eq 23 is a hybrid between the square-root dependence when θ f 0 and the Lagergren dependence when θ f θe. This is because

lim tf0

dθ 1 ) KMθe2 dt θ

(25)

When compared to eq 21 this would imply that in the limit θ f 0 the coefficient KM has the following interpretation

KM )

Klscin θe

(26)

Figure 1. The comparison of the theoretical kinetic adsorption isotherms calculated from the SRT equation (7) for three different values of the equilibrium surface coverage: θe ) 0.25 ()), θe ) 0.5 (0), and θe ) 0.75 (O) with their approximate values calculated from the MPFO equation (24) (s). In the case of panel (B) the linear representation, suggested by eq 24, was used. All calculations were performed for λ ) 1.1θe.

Then in the limit θ f θe, from eq 23 we obtain

lim

θfθe

dθ ) KM(θe - θ) + O[θe - θ]2 dt

(27)

By comparison with eq 10 in the limit θ f θe, the interpretation of KM is identical with that of k1 coefficient given by eq 13. The MPFO kinetic equation offers a very attractive simple way of correlating experimental data. Now we are going to show that this hybrid between the square-root and the Lagergren dependence simulates fairly well the behavior of the fully exact function θ(t) in eq 7 also in the mediate region between short and long adsorption times. We performed some simple model investigations, plotting kinetic isotherms calculated from eqs 7 and 24 as functions of the reduced time τ/τ0.9, where τ0.9 is the time when the surface coverage θ reaches the value θ ) 0.9θe. The dimensionless time for the MPFO equation is defined as τ ) KMt. The results are shown in Figures 1 and 2. Looking through Figures 1 and 2, we can see a quite good agreement between the theoretical kinetics isotherms predicted by the MPFO equation (24) and the SRT expression (7). The accuracy of MPFO equation (24) (treated as an approximation of the exact equation (7)) is increasing while the λ parameter is increasing. Then we have also found that when the value of λ is very high in comparison to θe, one can observe the reverse behavior, i.e., a decreasing accuracy of approximation (24). However, the situation that λ . θe is highly nontypical when considering the real kinetic experiments. This is because such high values of λ parameters mean that no significant changes in the solute concentration will be observed in the course of experiment. From the point of view of a successful application of an adsorption process, the highly desirable situation is that λ ≈ θe, i.e., when almost all the sorbate is removed from the solution. An inspection into literature seems to suggest that the most typical cases are these when λ lies between these two values λ ) 1.1θe and λ ) 2θe, accepted in our model calculations presented in Figures 1 and 2. A general conclusion can also be

Figure 2. The comparison of the theoretical kinetic adsorption isotherms calculated from the SRT equation (7) and from the MPFO equation (24) (s). All calculations were performed for λ ) 2θe. All the notations are the same as in Figure 1.

drawn that the MPFO equation reproduces the SRT expression (7) best for the low values of θe. Then, we also decided to compare the behavior of theoretical kinetic isotherms calculated from the full form of the IDM equation (14) with those calculated from the MPFO equation (24). Now, the dimensionless time for the IDM equation is defined as τ ) Dπ2t/r2. The results are shown in Figure 3. One can see there a somewhat systematically worse agreement than that in Figure 2. It seems, however, that the MPFO empirical equation

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Figure 4. The experimental kinetic isotherms for the RY/F-400 system reported by Yang and Al-Duri, measured at the four initial solute concentrations: 35.4 mg/L (O), 60.3 mg/L (0), 87.2 mg/L (4), and 131 mg/L ()). The solid lines (s) are the linear fits with abscissa values set as zero.

Figure 3. The comparison of the theoretical kinetic adsorption isotherms calculated from the IDM equation (14) and from the MPFO equation (24) (s). All the notations are the same as in Figure 1.

may simulate also the behavior of the kinetics governed by intraparticle diffusion. The general conclusion, therefore, is that distinguishing between these two kinetic mechanisms will be a difficult problem.

Analysis of Experimental Data In this section we introduce the following additional notation: N(t) denotes the amount adsorbed as a function of time, while Ne is the amount adsorbed at equilibrium. As an example we present here the analysis of the experimental data published in 2001 by Yang and Al-Duri10 on sorption of reactive dyes onto Filtrasorb-400 activated carbon. All details related to the experimental procedures and the adsorption system description can be found in their original paper. We have chosen the reactive yellow/activated carbon (RY/F-400) adsorption system for our further analysis. The adsorption kinetics for this system was studied at four initial solute concentrations, as can bee seen in Figure 5 in the original paper. The kinetic experiments were accompanied by the measurements of the equilibrium adsorption isotherm. Also, the value of monolayer capacity was determined, which allows presentation of Ne values as θe. Yang and Al-Duri found that the empirical Fritz-Schluender isotherm equation correlates best the equilibrium adsorption data. The determined parameters of this equation as well as the description of the technical conditions of the investigated system at which the experiments was carried out (mass of sorbent, volume of solution and initial sorbate concentrations) allow us to (1) calculate the Ne value corresponding to a given initial solute concentration and (2) transform the measured data into the N(t) vs t form. Then it is to be emphasized that the kinetic and the equilibrium adsorption isotherms were measured for different ranges of surface coverage. The Fritz-Schluender empirical equation was used to generate the values of the equilibrium adsorption isotherm in the region of the surface coverage at which the kinetics was experimentally monitored. But then we assumed, of course, that the Langmuir model can be accepted in the range of surface coverages where the sorption kinetics were measured. (10) Yang, X.; Al-Duri, B. Chem. Eng. J. 2001, 83, 15.

Figure 5. The tangents of the linear plots presented in Figure 4, plotted in the coordinates suggested by the IDM equation (18) (b) or by the SRT equation (22) (9) to describe adsorption kinetics in the initial region of surface coverages.

Both the SRT approach and the IDM equation (18) predict that at the initial short times the linear dependence of amount adsorbed on the square root of time is to be observed. Thus, in the first step of analysis, we fitted the initial part of measured kinetic data by the linear through origin function which corresponds to the SRT equation (22) and the IDM simplified equation (18). We took into account only those data points for which the relation N(t) e Ne/3 was fulfilled, as, according to the simple model investigation, this is the limit of the applicability of eq 18. The results are shown in Figure 4. Let us denote the tangents of these linear fits, presented in Figure 4, as R. According to the theories presented in the previous section, the values of these tangents should be (1) either a linear function of Ne, suggested by the IDM and eq 18 or (2) a linear function of (cinNe)1/2, suggested by the SRT equation (22). In both cases the values of the abscissas of these linear fits should be again set as zero, which is due to the physical meaning and the mathematical form of eqs 18 and 22. Figure 5 shows the results of our investigation. As one can see in Figure 5, the proposed analysis can hardly answer the question whether the SRT approach or IDM is more appropriate to represent the initial sorption kinetics. Therefore, in the next step we used full forms of both the IDM equation (14) and the SRT equation (5) to fit all the experimental data points, by adjusting the parameters Dπ2/r2 and Kls, respectively. The Langmuir isotherm equation (4) was used to calculate the values of KL for each pair of (ce, θe). The values of these best-fit parameters are collected in Table 1, and the goodness of the fits is presented in Figure 6. Looking into Figure 6 one may conclude that both IDM and SRT kinetic models offer a similar fit of experimental data. But

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Table 1. The Values of the Best-Fit Parameters Used to Fit the Kinetic Isotherms Measured by Yang and Al-Duria cin (mg/L)

Ne (mg)

Kls (L/(min/mg))

Dπ2/r2 (1/min)

Nmax/Ne

35.4 60.3 87.2 131

83.9 145.2 211.6 319.3

3.70 × 10-5 3.08 × 10-5 2.51 × 10-5 2.59 × 10-5

2.55 × 10-3 1.57 × 10-3 9.40 × 10-3 7.89 × 10-3

0.99 0.98 0.89 0.86

a The obtained agreement between theory and experiment is shown in Figure 6. Nmax is the maximum value of N(t) monitored during the experiment.

Figure 7. The MPFO linear representation (24) of the experimental kinetic data and their SRT approximation shown in Figure 6 and obtained by using the Kls parameters collected in Table 1. All notations as in Figure 4.

Figure 6. The agreement between the experimental kinetic isotherms and the theoretical isotherms calculated from the IDM equation (14) and the SRT equation (5) by using the best-fit parameters collected in Table 1. All notations as in Figure 4.

then the inspection of Table 1 seems to advocate for the SRT kinetic model. Namely, the Kls parameter values are less dependent on cin than those of the Dπ2/r2 parameter. Strong changes in the values of the parameters which should be constant do not speak in favor of an underlying physical model. In particular, very close to each other are the two values Kls ) 2.51 × 10-5 L/(mg/ min) and Kls ) 2.59 × 10-5 L/(mg/min), corresponding to the two highest initial concentrations: cin ) 87.2 mg/L and cin ) 131 mg/L. So, it seems important to notice that these two kinetic isotherms lie in a region which is further from equilibrium than the two others. This is depicted by the values of Nmax/Ne collected in Table 1. Therefore, one may conclude that the two kinetic isotherms corresponding to the two highest initial concentrations lie in the kinetic region where the SRT model applies better. The other two kinetic isotherms lie in a region which is closer to equilibrium, so they may be also influenced by another kinetic mechanism which is typical of the systems which are close to equilibrium. That kinetic mechanism has been identified in our previous paper as the one controlled by the rate of intraparticle diffusion.5 Thus, also in the case of the adsorption system investigated here by us, we arrive at the same picture of adsorption kinetics. In the initial region, far from equilibrium, the kinetics is governed by the rate of surface reactions. When the system approaches equilibrium, switching takes place from the surface reaction kinetics to that governed by the intraparticle diffusion. Now we are going to show that the MPFO linear representation (24) may serve as a test more sensitive to the changing kinetic mechanism than the usual N(t) vs t representation. Indeed, looking

Figure 8. (A) The MPFO linear representation (24) of those experimental data points for which the condition N(t) < 0.66 Ne is fulfilled and (B) the Lagergren plots (15) for the second parts of kinetic isotherms for which N(t) > 0.66 Ne. All notations as in Figure 4.

into Figure 6, it is difficult to judge whether or when switching takes place from one kinetic mechanism to another. A different picture emerges on the MPFO linear plot ln(Ne N(t)) + N(t)/Ne vs t, as shown in Figure 7. However, there is also another important fact relating to the presented MPFO linear representations of experimental data. Namely, these representations, presented in Figure 7 were obtained by using the experimentally determined (i.e., by using the parameters of Fritz-Schluender isotherm) Ne values. A much better linearity than those presented in Figure 7 can be obtained when treating Ne as the best-fit parameter. Nevertheless, accepting such a procedure in the case of the adsorption system investigated here leads us to higher Ne values than those obtained from the equilibrium adsorption isotherm. These higher Ne values were next used by us to calculate the corresponding ce values. It appeared that this procedure led us to unphysical values of equilibrium solute concentration: ce < 0. Thus, the conclusion can be drawn that only experimentally determined values of Ne should be used when the data transformation N(t) f ln(Ne N(t)) + N(t)/Ne is applied. Looking into Figure 7 one can observe a highly linear trend in the behavior of the experimental data from very short times, up to those corresponding to roughly two-thirds of the equilibrium surface coverage. This would mean that this is the coverage at which switching takes place from the surface reaction to the

The Modified Pseudo-First-Order Kinetic Equation

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Table 2. The Values of the Tangents (KM, Dπ2/r2) and of the Abscissas of Their Linear Plots (Data Regressions) Presented in Figure 8 MPFO plot

Lagergren plot

cin (mg/L)

ln Ne (SRT)

ln Ne + ln(6/π ) (IDM)

KM (1/min)

abscissa (exptl)

Dπ /r (1/min)

abscissa (exptl)

35.4 60.3 87.2 131

4.43 4.98 5.35 5.84

3.93 4.48 4.86 5.34

1.81 × 10-3 9.40 × 10-3 6.15 × 10-3 5.30 × 10-3

4.45 4.99 5.36 5.78

2.78 × 10-3 2.23 × 10-3 1.10 × 10-3 9.54 × 10-3

3.87 4.97 5.00 5.49

2

diffusional kinetic mechanism. We have faced a similar situation in our previous works, where such switching occurred at about 80% of the equilibrium surface coverage in the systems investigated by us.5 Then, as the remaining parts of the kinetic isotherms are closer to equilibrium, they should be well correlated by the Lagergren plots (15). Thus, assuming that switching in our present system occurs at 66% of the equilibrium surface coverage, we decided to construct the following graphical illustration. Every kinetic isotherm was divided into two parts: the first part containing N(t) values smaller than 0.66Ne and the second part the values N(t) > 0.66Ne. The first part was correlated by using the linear MPFO representation (24) whereas the second part was correlated by the linear Lagergren plot (15). Looking into Figure 8, we can see fairly good linear correlations, predicted by the assumption of switching from the surface reaction to the diffusional sorption kinetics. So, it seems interesting to analyze the values of the coefficients (tangents, abscissas) of their linear regressions. They are collected in Table 2. One can see when looking into Table 2 that the values of abscissas in the MPFO plots are in very good agreement with their theoretically predicted values, i.e., with ln Ne. However, in the case of the Lagergren plots, their abscissas are generally not equal to ln Ne as suggested by eq 11. Moreover, their values are not equal to ln Ne + ln (6/π2) as predicted by the diffusional Lagergren-like expression (15). This is more evidence for the fact that switching takes place from one kinetic mechanism to another. An explanation as to why such switching takes place may not be a trivial problem. At present we may launch the following hypothesis. There may exist a gradient of concentration at the entrance to pores of the adsorbent. So, in general the SRT equations should

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be combined yet with a kind of a film resistance equation. Such still complicated model of adsorption kinetics seems to be suggested by the data presented in Figure 4. There, best linear regressions will not originate at zero. Combining the SRT equation with a film resistance model would result into a system of two differential equations. We will present a more complicated model in a future publication.

Conclusions The MPFO equation can reproduce very well the behavior of the kinetic eqs 5, 7, 11, and 22 based on the SRT approach by assuming the Langmuir model of adsorption and a homogeneous solid surface. The accuracy of this approximation is the highest for the values of λ parameter, corresponding to the most common experimental conditions. Thus, the empirical MPFO equation can be treated as an approximation of our exact SRT equation. The linear representation of the MPFO equation (24) can be applied only when the true experimental values of Ne are known. This is because treating Ne as the best-fit parameter, leading to the best linearity of ln(Ne - N(t)) + N(t)/Ne vs t plots, enables obtaining unphysical values of Ne as well as ce when the mass balance relation is used. One can propose a simple method, based on the analysis of kinetic adsorption isotherms measured at very short, initial times of experiment, to distinguish between the SRT and the IDM kinetic models. This, however, would require measurements carried out with an exceptionally high accuracy. Such possibility is offered by applying the linear MPFO representation (24). This has led us to discover “switching” from the surface reaction to diffusional kinetics when the surface coverage reaches about two-thirds of the equilibrium surface coverage. LA8000448