Fractal-Like Kinetics for Adsorption on Heterogeneous Solid Surfaces

Dec 30, 2013 - to describe a physical meaning for the fractal-like kinetics in heterogeneous surface systems as well as that in homogeneous surfaces. ...
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Fractal-Like Kinetics for Adsorption on Heterogeneous Solid Surfaces Monireh Haerifar and Saeid Azizian* Department of Physical Chemistry, Faculty of Chemistry, Bu-Ali Sina University, Hamedan 65174, Iran ABSTRACT: A physical meaning of the fractal-like concept has been represented for kinetics of adsorption on energetically heterogeneous solid surfaces. This study shows that the adsorption kinetics at the solid/solution interface in a real system with different types of surface sites and then with different affinity for adsorption can be described by a fractallike approach. On the basis of this study, the history of process can affect the rate of process, and therefore the observed adsorption rate coefficient is a function of time. Employing this concept to extend some kinetic models leads to obtaining different fractallike kinetic models that can be utilized to simulate adsorption kinetics in systems with heterogeneous surfaces. The simulation of experimental kinetic data with fractal-like equations for the heterogeneous surfaces and the fact that their equilibrium data follow the Langmuir−Freundlich (Sips) isotherm indicate the applicability of these equations for real systems. The obtained results show that the observed adsorption rate coefficient in heterogeneous surface systems depends on time, and it decreases by passing time.

1. INTRODUCTION According to the literature, there are different adsorption kinetic models that can be employed by researchers to simulate kinetic data in solid/solution systems. Pseudo first-order (PFO)1 and pseudo second-order (PSO)2 models are the most popular ones. Also, the mixed 1,2-order equation (MOE)3 was introduced recently as a combination of the pseudo firstorder and pseudo second-order equations. The Langmuir kinetic model4 is one of the first theoretical models that was presented for systems that have a homogeneous solid surface. Moreover, the integrated kinetic Langmuir equation (IKL)5 is essentially derived from the classic Langmuir model. The appearance structure of this equation is very similar to the mixed 1,2-order equation (MOE); however, they have been derived by different concepts and foundations. On the other hand, the Langmuir−Freundlich kinetic model6 is an equation for adsorption in systems with heterogeneous solid surfaces, but it has no exact analytical solution. The statistical rate theory (SRT)7 is one of the new approaches for adsorption kinetic modeling at the solid/solution and gas/solid interfaces. This model has been theoretically solved for systems with homogeneous surfaces at initial times of process and close to equilibrium.8 Recently, we introduced the fractal-like concept in an adsorption system at the solid/solution interface.9 In this study, we focused on a typical homogeneous solid surface system to find a conceptual physical meaning of the fractal-like kinetics, and then we presented the time dependency of the rate coefficient in these systems to obtain the fractal-like kinetic models for adsorption at the solid/solution systems.9 Most of the adsorbents have a heterogeneous surface, so the presentation of kinetic models for these systems is very important from a practical point of view. Now, we are going to extend and generalize our previous study9 for adsorption on an energetically heterogeneous solid surface system. So, we will try © 2013 American Chemical Society

to describe a physical meaning for the fractal-like kinetics in heterogeneous surface systems as well as that in homogeneous surfaces. Also, we will utilize this concept in different sorption kinetic models that can be useful to simulate kinetic data in systems with energetically heterogeneous solid surfaces. In some of the experimental adsorption kinetic data, the rate of adsorption not only varies with fluid-phase concentration but also exhibits a time-dependent rate constant. The models presented based on a fractal-like concept can explain and fit such data.

2. FRACTAL-LIKE ADSORPTION KINETICS IN HOMOGENEOUS SOLID SURFACE SYSTEMS In our previous study, we introduced a new kinetic model for adsorption at the solid/solution interface by using a fractal-like concept.9 In fact, we considered that this process occurs in systems whose solid surfaces are homogeneous. So, we used a schematic pattern such as Figure 1 to represent a physical meaning of the fractal-like kinetics. As it is clear, in this figure the surface sites have been considered identical and uniform. Also, we indicated that by passing time different paths may appear for molecules to be adsorbed on the surface sites; therefore, there is only one possible path at initial times of the process, and then the number of adsorption paths increases with time (these paths have been indicated by 0, A, and B). The adsorption rate constants are different for different paths, where ka,0 > ka,A > ka,B > ....9 By definition of the weight factor of each path (wi (i = 0, A, and B)) we considered the contribution of each rate constant to the observed adsorption rate coefficient at any time (ka,obs). So, it is equal to9 Received: November 11, 2013 Revised: December 8, 2013 Published: December 30, 2013 1129

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Figure 1. Schematic plan of the adsorption process as a function of time in homogeneous surfaces. 0, A, and B are the possible paths for adsorption of solute particles on surface sites.

Figure 2. Schematic representation of the adsorption process as a function of time in heterogeneous solid surfaces. 0*, A*, and B* are three types of surface sites for adsorption of particles. z

ka,obs =

z

∑ wki a,i ,

∑ wi = 1,

i=0

i=0

Since the contribution of smaller rate constants increases by passing time, we concluded that the observed rate coefficient (ka,obs) decreases as the time passes. Therefore, we showed that

i = 0, A, B, ..., Z (1) 1130

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Table 1. Fractal-Like Kinetic Models Based on PFO, PSO, MOE, and Exponential Equations and Equation 8a adsorption kinetic models

kinetic equation

1,12

q = qe[1 − exp(− k1t )]

PFO

PSO2,12

q=

q = qe

MOE3

fractal-like adsorption kinetic models F−L PFO

k 2qe2t

F−L PSO9

1 + k 2qet

1 − exp(− k1t ) 1 − f2 exp(− k1t )

q=

F−L MOE9

′ t )] q = qe ln[2.72 − 1.72 exp(− kExp

exponential13

F−L kinetic equation

′ t α)] q = qe[1 − exp(− k1,0

9

q = qe

′ qe2t α k 2,0 ′ qet α 1 + k 2,0

′ t α) 1 − exp(− k1,0 ′ t α) 1 − f2 exp(− k1,0

″ t α)] q = qe ln[2.72 − 1.72 exp(− kExp,0

F−L exponential13

a In these equations, k1, k2, and kExp ′ are PFO, PSO, and exponential adsorption rate coefficients; also, k1,0 ′ , k2,0 ′ and kExp,0 ″ are the rate coefficients of F− ′ (n = 1,2) = kn,0/α, kExp,0 ″ = kExp,0 ′ /α, and α = 1 − h. L PFO, F−L PSO, and F−L exponential equations, where: kn,0

where w*0 and w*A are the weight factor for rate constants of adsorption on the 0* and A* sites, respectively. At this stage w*0 + wA* = 1. At later times, t3 (where t3 > t2), most of the 0* and some of the A* sites were occupied. Therefore, solute particles starts to adsorb on the third site (B*) with rate constant of k*a,B in addition to 0* and A* (Figure 2c). In other words, at this time, there are three types of surface sites for adsorption but with different probabilities. In general, k*a,0 > k*a,A > k*a,B. So, the observed rate constant for t3 is equal to

the observed rate coefficient is not a constant parameter, and it decreases by passing time as a fractal-like or time dependency parameter.9 According to the fractal-like approach which was first proposed by Kopelman10 for the rate coefficient in homogeneous reactions and the above explanations, the observed adsorption rate coefficient was introduced as follows9 ka,obs = ka,0t −h ,

(0 ≤ h ≤ 1),

(t ≥ 1)

(2)

where h is a constant parameter.

3. FRACTAL-LIKE ADSORPTION KINETICS IN HETEROGENEOUS SOLID SURFACE SYSTEMS In this section we are going to introduce a physical meaning of the fractal-like concept for adsorption on heterogeneous surfaces at solid/solution systems. For this purpose, we consider typically three kinds of active sites for adsorption on a solid surface. These sites have been specified by 0*, A*, and B* in Figure 2. Then, let us consider that the activation energy for adsorption on the mentioned sites are Ea,0 * , Ea,A * and Ea,B *, respectively, and also E*a,0 < E*a,A < E*a,B. The relationship between rate constant and the activation energy of adsorption is given by the Arrhenius equation11 ⎛ E* ⎞ a, j ⎟ ka,*j = A′ exp⎜⎜ − ⎟ RT ⎝ ⎠

* = w0*ka,0 * + wA*ka,A * + wB*ka,B * ka,obs

(j = 0*, A*, B*, ..., Z*) (3)

(for initial time)

z*

* = ka,obs

(for t 2 > t1)

∑ j=0*

z

w*j ka,*j ,

∑ w*j = 1 j=0

(j = 0*, A*, B*, ..., Z*)

(7)

It is necessary to note that at initial times of the process k*a,obs is very large because w*0 ≈ 1 and the contribution of k*a,0 is large. By passing of time, the contribution of ka,0 * decreases, while the contribution of ka,A * and then ka,B * increases. In fact, this conclusion is similar to that we explained before in the case of homogeneous surfaces (section 2).9 As mentioned before, the weight factors, wj*, change with time, and the rate coefficient of adsorption is a time-dependent coefficient. Since by passing of time the w*0 value decreases while w*B increases, also k*a,0 > k*a,A > k*a,B, and by considering eq 6, one can conclude that the observed rate coefficient of adsorption, ka,obs * , decreases with time. Therefore, on the basis of the Kopelman fractal-like approach for the rate coefficient9,10 and also eq 2 we can write the following expression to introduce the fractal-like observed rate coefficient for adsorption on heterogeneous surfaces.

(4)

In the next moments, t2, most of the surface sites of type 0* are occupied by adsorbed molecules as has been shown in Figure 2b. At this step, sorbates can be adsorbed on both 0* and A* with the rate constants k*a,0 and k*a,A, respectively, where ka,0 * > ka,A * , and in this case the observed rate coefficient can be defined as * = w0*ka,0 * + wA*ka,A * ka,obs

(6)

where w0* + wA* + wB* = 1. It is very important to note that the weight factors are a function of time, and their values change with time. For example, at initial times of adsorption w*0 is close to unity, but wB* is close to zero; however, at longer times the w0* value goes toward zero, but the wB* value increases. Therefore, it can be concluded that the adsorption pathway changes with time from sites with higher adsorption energy to the sites with lower adsorption energy. Therefore, according to the above explanations we can write the following relationship between the observed adsorption rate coefficient (k*a,obs) and the rate constants (k*a,j) of adsorption onto different surface sites (0*, A*, B*, ..., Z*) of a heterogeneous solid surface, as

where ka,j * represents the adsorption rate constant; A′ is the Arrhenius factor; R is the gas constant; E*a,j is the activation energy of adsorption for the adsorption on site j; and T is the temperature. This equation indicates that at constant temperature the lower value of activation energy leads to higher adsorption rate constant. Therefore, we can conclude that at initial times of sorption, when all the adsorption sites on the surface are empty, particles prefer to adsorb only on 0* sites where the rate constant for this site is k*a,0(Figure 2a). So, the observed adsorption rate coefficient (k*a,obs) at initial times of the process is equal to * = ka,0 * ka,obs

(for t3 > t 2)

(5) 1131

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Table 2. Fitted Constants of Fractal-Like Kinetic Models at Different Initial Concentrations of Solute for Adsorption of Methylene Blue (MB) on Carbon Nanotubes16 and Also Adsorption of Ammonium (NH4+) onto Clinoptilolite17 system (kinetic model)

C0 (mg/L)

qe (mg/g)

kn,0 * (n = 1 and 2)

α

f2

R2

MB/carbon nanotubes (F−L PSO)

20 30 40 160 250

141.0 171.6 195.8 11.2 15.2

0.001 0.0008 0.0009 0.05 0.14

0.92 0.85 0.68 0.83 0.58

0.752 0.097

0.999 0.999 0.996 0.994 0.994

NH4+/clinoptilolite (F−L MOE)

Figure 3. Changes of q versus time for (a) adsorption of methylene blue onto carbon nanotubes16 and (b) adsorption of ammonium from aqueous solution onto clinoptilolite17 at different initial concentrations of adsorbate. The symbols and lines are the experimental and the calculated values based on the fractal-like kinetic models.

* = ka,0 * t −h , ka,obs

(0 ≤ h ≤ 1),

(t ≥ 1)

Therefore, we can employ eq 8 to extend kinetic models such as pseudo first-order,1,12 pseudo second-order,2,12 mixed 1,2order,3 and also exponential13 adsorption kinetic models. Table 1 lists different fractal-like kinetic models which can be used to describe adsorption kinetics at the solid/solution interface, where the solid surface is heterogeneous. In these equations, q and qe are defined as the amount of adsorbate per unit mass of sorbent at time t and at equilibrium, respectively. Also, kn, kn,0 ′

(8)

This equation shows the time dependency of the observed rate coefficient of adsorption so that it decreases as the time increases. The time dependency of the rate coefficient was described by a fractal-like approach.9 In other words, this time dependency means that the history of process can affect the rate coefficient and then the rate of process.9,10 1132

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Figure 4. Variation of kn,obs * = kn,0 * t−h (n = 1, 2) with time based on eq 8 for (a) adsorption of methylene blue onto carbon nanotubes16 and (b) adsorption of ammonium onto clinoptilolite17 at different initial concentrations of adsorbate.

(n = 1,2), kExp ′ , and kExp,0 ″ are the kinetic rate coefficients, and α (where α = 1 − h) is the fractional time index. The fractal-like kinetic models which are listed in Table 1 for heterogeneous surfaces are similar to those introduced for homogeneous surfaces.9 So, the main question that should be answered is: ″is it possible to discriminate homogeneous and heterogeneous surfaces by fitting of adsorption kinetic data to the fractal-like kinetic models?″. The answer is: No. So, at first we have to distinguish the homogeneous and heterogeneous surface by use of equilibrium data; for example, if the adsorption equilibrium data follow the Langmuir isotherm4,14 then the surface is homogeneous, while if they follow the Langmuir−Freundlich (Sips) equation,6,14,15 the surface is heterogeneous. Now, following of adsorption kinetic data with fractal-like models for homogeneous surfaces causes the presence of different paths for adsorption.9 If the surface is heterogeneous (based on the results of the equilibrium isotherm model) and the kinetic data follow the fractal-like kinetic model, one can conclude that there are different

adsorption sites with different affinity for adsorption. So, by the passing of time, the sites of adsorption change, and therefore the observed rate coefficient is dependent on the time.

4. RESULT AND DISCUSSION To investigate the applicability of the fractal-like kinetics and also time dependency effect on the adsorption rate coefficient for sorption systems with heterogeneous solid surfaces, we use these models to simulate experimental kinetic data for adsorption at the solid/solution interface. For this purpose, we use two experimental systems including adsorption of methylene blue (MB) onto carbon nanotubes16 and also removal of ammonium (NH4+) from aqueous solutions by using clinoptilolite.17 On the basis of the studies that were conducted on the mentioned systems, the equilibrium kinetic data were successfully fitted by the Langmuir−Freunlich isotherm (for the first system: n′ = 1.33, R2 = 0.999;16 and also for the latter: n′ = 1.82, R2 = 0.99117). So, we applied the fractal-like kinetic models (Table 1) to analyze kinetic data at 1133

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(4) Langmuir, I. The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum. J. Am. Chem. Soc. 1918, 40, 1361−1403. (5) Marczewski, A. W. Analysis of Kinetic Langmuir Model. Part I: Integrated Kinetic Langmuir Equation (IKL): A New Complete Analytical Solution of the Langmuir Rate Equation. Langmuir 2010, 26, 15229−15238. (6) Cheung, C. W.; Porter, J. F.; McKay, G. Sorption Kinetics for the Removal of Copper and Zinc from Effluents Using Bone Char. Sep. Purif. Technol. 2000, 19, 55−64. (7) Rudzinski, W.; Plazinski, W. 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. J. Phys. Chem. C 2007, 111, 15100−15110. (8) Azizian, S.; Bashiri, H. Adsorption Kinetics at the Solid/Solution Interface: Statistical Rate Theory at Initial Times of Adsorption and Close to Equilibrium. Langmuir 2008, 24, 11669−11676. (9) Haerifar, M.; Azizian, S. Fractal-Like Adsorption Kinetics at the Solid/Solution Interface. J. Phys. Chem. C 2012, 116, 13111−13119. (10) Kopelman, R. Fractal Reaction Kinetics. Science 1988, 241, 1620−1626. (11) Hamdaoui, O.; Saoudi, F.; Chiha, M.; Naffrechoux, E. Sorption of Malachite Green by a Novel Sorbent, Dead Leaves of Plane Tree: Equilibrium and Kinetic Modeling. Chem. Eng. J. 2008, 143, 73−84. (12) Azizian, S. Kinetic Models of Sorption: A Theoretical Analysis. J. Colloid Interface Sci. 2004, 276, 47−52. (13) Haerifar, M.; Azizian, S. An Exponential Kinetic Model for Adsorption at Solid/Solution Interface. Chem. Eng. J. 2013, 215−216, 65−71. (14) Saha, B.; Orvig, C. Biosorbents for Hexavalent Chromium Elimination from Industrial and Municipal Effluents. Coord. Chem. Rev. 2010, 254, 2959−2972. (15) Azizian, S.; Haerifar, M.; Basiri-Parsa, J. Extended Geometric Method: A Simple Approach to Derive Adsorption Rate Constants of Langmuir-Freundlich Kinetics. Chemosphere 2007, 68, 2040−2046. (16) Shahryari, Z.; Goharrizi, A. S.; Azadi, M. Experimental Study of Methylene Blue Adsorption from Aqueous Solutions onto Carbon Nano Tubes. Int. J. Water Res. Environ. Eng. 2010, 2, 016−028. (17) Tosun, I.̇ Ammonium Removal from Aqueous Solutions by Clinoptilolite: Determination of Isotherm and Thermodynamic Parameters and Comparison of Kinetics by the Double Exponential Model and Conventional Kinetic Models. Int. J. Environ. Res. Public Health 2012, 9, 970−984.

various initial concentrations of adsorbates. Accordingly, we found that the kinetic data can be simulated by F−L PSO and F−L MOE models very well for ″MB/carbon nanotubes″ and ″NH4+/clinoptilolite″ sorption systems, respectively. The calculated parameters and the obtained results are listed in Table 2. Also, the changes of q versus time have been plotted for different initial concentrations of adsorbate in Figure 3. The results and the plots of Figure 3 indicate that the fractal-like kinetic models can fit experimental data very well. It is clear that by increasing the amount of initial concentration of solute the value of α decreases because the possibility for adsorption onto other sites increases. Moreover, to study the time dependency of the adsorption rate coefficients for the experimental mentioned systems with heterogeneous surfaces, we plotted the changes of the observed adsorption rate coefficients (kn,obs * = kn,0 * t−h, n = 1, 2) by time in Figure 4. This figure obviously shows that the observed rate coefficient is not a constant parameter, and it decreases as the time increases.

5. CONCLUSION In this study we presented a physical meaning for the fractallike approach for adsorption kinetics on energetically heterogeneous surfaces. We indicated that the adsorbate particles may choose various types of surface sites with different weight factors and also various rate constants for adsorption. In fact, by passing time, the probability of adsorption on sites with a lower value of rate constant is increased. So, it was concluded that the observed adsorption rate coefficient depends on time by eq 8. Also, based on the fractal-like concept, the fractal-like equations can be useful to analyze kinetic data for adsorption systems with heterogeneous solid surfaces. Furthermore, using the fractal-like kinetic models to describe experimental data points at various initial concentrations of adsorbate shows that they can simulate kinetic data very well. In these systems the surface of adsorbents according to reported results was heterogeneous. As a result, it was indicated that the observed adsorption rate coefficient in heterogeneous surface systems is not a constant parameter, and it depends on time so that it may decrease as the time increases. In other words, the history of the process can affect the rate coefficient of adsorption.

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AUTHOR INFORMATION

Corresponding Author

*Fax: +98-811-8380709. E-mail: [email protected] or [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the financial support of Bu Ali Sina University.

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REFERENCES

(1) Lagergren, S. About the Theory of So-Called Adsorption of Soluble Substances. K. Sven. Vetenskapsakad. Handl. 1898, 24, 1−39. (2) Ho, Y. S.; McKay, G. The Kinetics of Sorption of Divalent Metal Ions onto Sphagnum Moss Peat. Water Res. 2000, 34, 735−742. (3) Marczewski, A. W. Application of Mixed Order Rate Equations to Adsorption of Methylene Blue on Mesoporous Carbons. Appl. Surf. Sci. 2010, 256, 4145−5152. 1134

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