Solution Interface

Ion exchange models include the homogeneous diffusion model (HDM) based on Fickʼs law .... Figure 1a shows the generated kinetic data (t, q) of this ...
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Langmuir 2008, 24, 13013-13018

13013

Description of Desorption Kinetics at the Solid/Solution Interface Based on the Statistical Rate Theory Saeid Azizian* and Hadis Bashiri Department of Physical Chemistry, Faculty of Chemistry, Bu-Ali Sina UniVersity, Hamedan 65174, Iran ReceiVed September 10, 2008. ReVised Manuscript ReceiVed September 18, 2008 Desorption is one of the popular methods for the design and regeneration of catalysts. For better understanding and modeling of this process, it is important to have models with theoretical basis. In the present work, the statistical rate theory (SRT) approach was used for the description of desorption kinetics at the solid/solution interface. Based on the SRT approach, two rate equations at initial times of desorption have been derived. A comparison between these two rate equations was done based on numerically generated kinetic data points (t, q) by the SRT equation. On the basis of experimental data, it has been shown that the kinetics of desorption can be analyzed by the SRT rate equation. Also, the experimental data approve the accuracy of derived rate equations at initial times of desorption.

Introduction The sorption process plays an important role for reducing pollutants in natural and industrial systems. Adsorption and desorption are time dependent processes. Sometimes adsorption of contaminants occurs in aqueous solutions, whereas desorption of contaminants is commonly accomplished in organic solvents for effective regeneration.1-4 The modeling of kinetics of adsorption at the solid/solution interface has been studied extensively, but the kinetic modeling of desorption has been studied less. Therefore, the kinetic models that represent the time dependency of desorption process should be developed. There are different kinetic models that can be used to analyze the kinetics of adsorption and desorption processes. Some of the simple models of adsorption kinetics include the pseudo-first order model,5 pseudo-second order model,6,7 modified pseudofirst order model (MPFO),8 and so on. These kinetic models have been on the empirical basis until recently. In 2004 and based on the classical fundamental theory of actived adsorption/ desorption (TAAD) approach, Azizian derived pseudo-first order and pseudo-second order models for the first time.9 Rudzinski and Plazinski derived the pseudo-first order and the pseudosecond order models by applying the statistical rate theory (SRT) approach.10-12 Also Azizian developed the pseudo-second order model for adsorption onto two different sites.13 Recently, Azizian and co-workers presented an approximate method to obtain the rate constants of Langmuir-Freundlich kinetics.14,15 Most recently, we derived theoretically the MPFO kinetics model for the first time by applying the SRT approach.16 * To whom correspondence should be addressed. Fax: +98-811-8257404. Email: [email protected] and [email protected]. (1) Karimi-Jashni, A.; Narbaitz, R. M. Water Res. 1997, 31, 3039. (2) Mollah, A. H.; Robinson, C. W. Water Res. 1996, 30, 2907. (3) Goto, M.; Hayashi, N.; Goto, S. EnViron. Sci. Technol. 1986, 20, 463. (4) Chatzopoulos, D.; Varma, A.; Irvine, R. L. AIChE J. 1993, 39, 2027. (5) Lagergren, S. Kungliga SVenska Vetenskapsakademiens, Handlingar 1898, 24, 1. (6) Ho, Y. S.; McKay, G. Water Res. 2000, 34, 735. (7) Blanchard, G.; Maunaye, M.; Martin, G. Water Res. 1984, 18, 1501. (8) Yang, X.; Al-Duri, B. J. Colloid Interface Sci. 2005, 287, 25. (9) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (10) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514. (11) Rudzinski, W.; Plazinski, W. J. Phys. Chem. C 2007, 111, 15100. (12) Rudzinski, W.; Plazinski, W. Appl. Surf. Sci. 2007, 253, 5827. (13) Azizian, S. J. Colloid Intefrace. Sci. 2006, 302, 76. (14) Azizian, S.; Haerifar, M.; Basiri-parsa, J. Chemosphere 2007, 68, 2040. (15) Azizian, S.; Haerifar, M.; Bashiri, H. Chem. Eng. J., published online May 23, http://dx.doi.org/10.1016/j.cej.2008.05.024.

For modeling of desorption kinetics, different models were proposed.17-24 Ion exchange models17-21 and simple kinetic models22-24 are the most popular kinetic models which have been used to describe kinetics of desorption. The first order desorption model (Lagergren)22–24 as a function of the amount of adspecies is given by

q ) q0 exp(-KD1t)

(1)

where q is the amount of adsorbate at any time and q0 is the amount of adspecies at t ) 0. KD1 is the desorption rate constant for the first order model. The second order desorption model22,23 as a function of sorption loading is

(

q ) q0

1 β2 + KD2t

)

(2)

where β2 is a constant and KD2 is the desorption rate constant for the second order model. Ion exchange models include the homogeneous diffusion model (HDM) based on Fick’s law and the shrinking core model (SCM) based on modified Levenspiel’s theory.17-19 The mentioned simple first order and second order models of desorption kinetics were presented experimentally, without any theoretical basis. There are other kinetic models which have been applied for modeling of desorption kinetics.25-27 The purpose of the present work is to apply the new SRT approach for description of the kinetics of desorption at the solid/ solution interface. (16) Azizian, S.; Bashiri, H. Langmuir, 2008, 24, 11669. (17) Guibal, E.; Larkin, A.; Vincent, T.; Tobin, J. M. Ind. Eng. Chem. Res. 1999, 38, 4011. (18) Rao, M. G.; Gupta, A. K. Chem. Eng. J. 1982, 24, 181. (19) Juang, R. S.; Ju, C. Y. Ind. Eng. Chem. Res. 1998, 37, 3463. (20) Shih, Y.-H. Colloids Surf., A 2008, 317, 159. (21) Chen, Y.; Pawliszyn, J. Anal. Chem. 2004, 76, 5807. (22) Ho, Y. S.; Ng, J. C. Y.; McKay, G. Sep. Purif. Methods 2000, 29, 189. (23) Li, Z. Langmuir 1999, 15, 6438. (24) Kim, S.; Kim, Y.-K. Chem. Eng. J. 2004, 98, 237. (25) Purkait, M. K.; DasGupta, S.; De, S. J. EnViron. Manage. 2005, 76, 135. (26) Chang, Y.-K.; Chu, L.; Tsai, J.-C.; Chiu, S.-J. Process Biochem. 2006, 41, 1864. (27) Saffron, C. M.; Park, J.-H.; Dale, B. E.; Voice, C. T. J. EnViron. Technol. 2006, 40, 7662.

10.1021/la8029769 CCC: $40.75  2008 American Chemical Society Published on Web 10/25/2008

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Azizian and Bashiri

Theory The SRT approach was proposed by Ward based on quantum mechanics and thermodynamics.28,29 The SRT approach has been used for adsorption at gas/solid30 and solid/solution10-12,31-33 interfaces by Rudzinski et al. Recently, we developed the SRT approach for competitive adsorption at the solid/solution interface34 and also for adsorption kinetics at initial times of adsorption and also close to equilibrium.16 Based on the SRT approach, the rate of adsorption at the solid/solution interface is expressed by10,28

[ (

)

(

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

)]

θ - kT ln qs 1-θ

µb ) µb° + kT ln c

[

]

(6)

where KL is the Langmuir constant and is defined as

( )

KL ) qs exp

µb° kT

(7)

and the coefficient Kls′ can be written as11

Kls′ ) Klsce(1 - θe)

[

β(θi - θ)(1 - θ) dθ θ ) Kls′ KL dt θ KLβ(θi - θ)(1 - θ)

(8)

where ce is the equilibrium concentration of the solute and θe is the surface coverage at equilibrium. Hereafter, we want to apply the SRT for the modeling of desorption kinetics at the solid/ solution interface. Previously, Rudzinski et al. have been used the SRT for the modeling of desorption kinetics at the gas/solid interface.35-37 (28) Ward, C. A. J. Chem. Phys. 1977, 67, 229. (29) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (30) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (31) Rudzinski, W.; Plazinski, W. Langmuir 2008, 24, 6738. (32) Rudzinski, W.; Plazinski, W. Langmuir 2008, 24, 5393. (33) Panczyk, T.; Rudzinski, W. EnViron. Sci. Technol. 2008, 42, 2470. (34) Azizian, S.; Bashiri, H.; Iloukhani, H. J. Phys. Chem. C 2008, 112, 10251. (35) Panczyk, T.; Gac, W.; Panczyk, M.; Borowiecki, T.; Rudzinski, W. Langmuir 2006, 22, 6613. (36) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. AdV. Colloid Interface Sci. 2000, 84, 1. (37) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A.; Gac, W. Appl. Catal., A 2002, 224, 299.

]

(10)

Equation 10 is the basic equation of SRT for kinetics of desorption at the solid/solution interface. In this section, we want to derive a new and simple equation for desorption kinetics at initial times of desorption from eq 10. At initial times of desorption, the value of c is very low, so the adsorption rate (the first term in the RHS of eq 10) is ignorable. Therefore, at initial times of desorption, eq 10 simplifies to

(

dθ -θ ) Kls′ dt KLβ(θi - θ)(1 - θ)

)

(11)

Integration of eq 11 with the boundary condition θ(t ) 0) ) θi yields

θi ln θ - θ(1 + θi) +

( )

θi Kls′ θ2 ) θi ln θi - θi 1 + t 2 2 KLβ (12)

or

θi ln θ - θ(1 + θi) +

(5)

where c is concentration of solute in the bulk phase. By substitution of eqs 4 and 5 into eq 3

(9)

where β is a constant, θi is the fractional coverage of adspecies at the start of desorption process, and θ is the residual fractional coverage of adspecies on the surface. By substitution of eq 9 into eq 6, one arrives at

(4)

where qs is the molecular partition function of an adsorbed molecule. For solid/solution systems, the chemical potential of the adsorbate in the bulk phase (µb) is given by12

dθ c(1 - θ) θ 1 ) Kls′ KL dt θ KL c(1 - θ)

c ) β(θi - θ)

(3)

where µs and µb are the chemical potentials of the adsorbate on the surface and in the bulk phases, respectively. θ is the fractional surface coverage defined as θ ) q/qm (q is the amount of adsorbate, and qm is the maximum value of q). Kls′ is the adsorption/ desorption rate at equilibrium, k is the Boltzmann constant, and T is the absolute temperature. The first and second sentences in the brackets of the right-hand side (RHS) of eq 3 show the contribution of the adsorption (forward) and desorption (reward) processes, respectively. For the Langmuirian adsorption, the chemical potential of the adsorbate on the surface (µs) can be expressed by10

µs ) kT ln

The bulk concentration of solute (c) at the start of the desorption process is equal to zero, but it increases during desorption process as

θ2 ) R - Kt 2

(13)

where R is a constant and is equal to θi ln θi - θi(1 + θi/2). Thus, at initial times of desorption, the θi ln θ - θ(1 + θi) + θ2/2 term is a linear function of time with a slope of -K ) -Kls′/KLβ and intercept of R. Equation 12 (or 13) is a new and simple equation for the kinetics of desorption which can be applied for kinetic modeling at initial times of desorption. It is also possible to derive a more simple equation for desorption kinetics at very short initial times of desorption. When assuming that the desorption process is at the very short initial times, the θ/(1 - θ) term can be approximated as θi/(1 - θi), and therefore, from eq 11, the following expression can be obtained

(

-θi dθ ) Kls′ dt KLβ(θi - θ)(1 - θi)

)

(14)

Integration with the boundary condition θ(t ) 0) ) θi and then rearrangement gives

(

)

1 1

2K1s′θi 2 2 θ ) θi t KLβ(1 - θi)

(15)

or 1 ′ 2

θ ) θi - K t

(16)

where K′ is a constant and is equal to {[2Kls′θi]/[KLβ(1 - θi)]}1/2. Therefore, at very short initial times of desorption, θ is a linear function of t1/2. The plot of θ versus t1/2 will be a straight line with the slope of -K′ and an intercept of θi. A similar θ versus

Solid/Solution Interface Desorption Kinetics

Langmuir, Vol. 24, No. 22, 2008 13015 Table 1. Constants Used for Kinetic Modeling

system SRT (generated data) vitamin E/silica chromium(VI)/hydrotalcit

q0 (mg/g)

qm (mg/g)

Mw (g/mol)

cadsorbent (g/L)

60.00 53.86 × 10-1 61.94 × 10-4

12.00 × 10 36.21 × 10-3 17.00

60.00 43.07 × 101 52.00

0.10 2.00 × 101 1.00

1

Table 2. Values of ka and kd Used for Generation of Different Systems and Kls Values Obtained for Different Systems Based on Different Kinetic Models Kls system

ka (min )

kd (mol /L · min)

SRT

eq 12

eq 15

generated data 1 (by SRT) generated data 2 (by SRT) generated data 3 (by SRT) generated data 4 (by SRT) vitamin E/silica chromium(VI)/hydrotalcit

1.00 × 10-5 1.00 × 10-5 1.00 × 10-5 1.00 × 10-5 1.00 × 10-3 1.40 × 10-8

1.00 × 10-5 1.00 × 10-4 1.00 × 10-10 1.00 × 10-15 1.00 × 10-13 2.70 × 10-10

4.06 4.10 1.64 0.11 2.09 × 104 1.07 × 10-3

4.06 4.10 1.62 0.11 1.93 × 104 1.10 × 10-3

3.93 3.86 1.59 0.10 0.96 × 104 1.05 × 10-3

-1

2

t1/2 linear dependence has been predicted for adsorption where the surface reaction is the rate controlling step35 and also where the interparticle diffusion is the rate controlling step.38 It is important to note that in the case of adsorption the intercept is equal to zero but in the case of desorption the intercept is θi. Also, for adsorption the slope is positive, but for desorption the slope is negative. However, eq 15 (or 16) is a simple equation for initial times of desorption, but eq 12 (or 13) is more general than it. In the next section, with using both generated and experimental data, we will compare the applicability of these equations.

2

Figure 1b, eq 12 can be applied in a wide range of desorption times for these generated kinetic data points.

Result and Discussion In this section, the applicability of the derived equations at initial times of desorption (eqs 12 and 15) will be analyzed. For this purpose, four sets of hypothetical kinetic data points (t, q) were generated based on the SRT rate equation. For simplification of the SRT rate equation (eq 10), the following constants were defined:

ka ) Kls′KL kd )

Kls′ KL

(17) (18)

Substitution of these equations into eq 10 gives

β(θi - θ)(1 - θ) dθ θ ) ka - kd dt θ β(θi - θ)(1 - θ)

(19)

Stochastic numerical simulation was used for generation of kinetic data points by using the CKS package developed by Houle and Hinsberg.39 Recently, we utilized this method for numerical solution of some adsorption kinetics equations.15,16,34 The present simulation was performed by considering certain values of q0, qm, ka, and kd. Four different sets of kinetic data points (t, q) with similar initial conditions (Tables 1 and 2) but different kd values (Table 2) were generated. The different values of kd led to different values of the Langmuir equilibrium constant (KL). For the first system, KL is equal to unity (ka ) 1.00 × 10-5 min-1, kd ) 1.00 × 10-5 mol2/L2 · min). Figure 1a shows the generated kinetic data (t, q) of this system. The values of θi ln θ - θ(1 + θi) + (θ2/2) as a function of time (eq 12) were calculated and are presented in Figure 1b. This graph is a straight line with a high correlation coefficient (R2 ) 1). As shown in (38) Suzuki, M. Adsorption Engineering; Kodansha: Tokyo, 1990. (39) Houle, F. A.; Hinsberg, W. D. Surf. Sci. 1995, 368, 329.

Figure 1. (a) Numerically generated points (t, q) based on the SRT equation ka ) kd ) 1.00 × 10-5. (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

13016 Langmuir, Vol. 24, No. 22, 2008

Figure 2. (a) Numerically generated points (t, q) based on the SRT equation for ka ) 1.00 × 10-5 and kd ) 1.00 × 10-4. (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

The plot of q versus t1/2 (based on eq 15) is shown in Figure 1c. However, this linear plot has a high correlation coefficient (R2 ) 1), but it is clear that this equation (eq 15) can be applied in a shorter time range of desorption in comparison to eq 12. This means that eq 12 is more general than eq 15. For this set of generated kinetic data points based on the SRT equation, the value of Kls was calculated by eqs 8 and 17 (Table 2). Also, from the slopes of the linear plots in Figure 1b and c, the Kls values were calculated, and they are listed in Table 2. The obtained values of Kls based on eqs 12 and 15 are very close to the original value of Kls based on the SRT rate equation. This means that eqs 12 and 15 are suitable for analysis of kinetic data at initial times of desorption. The second set of kinetic data points were generated for ka < kd (ka ) 1.00 × 10-5 min-1, kd ) 1.00 × 10-4 mol2/L2 · min), and they are shown in Figure 2a. The values of θi ln θ - θ(1 + θi) + (θ2/2) versus time based on eq 12 were calculated, and they are presented in Figure 2b. As shown in this figure, the plot is completely linear in a wide range of times. The plot of q versus t1/2 (eq 15) is shown in Figure 2c. Although this plot is linear, its linearity is in a very narrow range of time. This means that eq 12 can be applied in the longer time range in comparison to

Azizian and Bashiri

Figure 3. (a) Numerically generated points (t, q) based on the SRT equation for ka ) 1.00 × 10-5 and kd ) 1.00 × 10-10. (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

eq 15. By using the tangents of the linear plots in Figure 2b and c, the Kls values were calculated (Table 2). Comparison of these Kls values with the original value from the SRT equation shows a good agreement between the Kls values. Two other sets of kinetic data points were generated for ka > kd (Table 2), and they are presented in Figures 3a and 4a. Figures 3b and 4b show the plot of θi ln θ - θ(1 + θi) + (θ2/2) versus time (based on eq 12) for these kinetic data points. The plots of q versus t1/2 (eq 15) are shown in Figures 3c and 4c. For both of these systems, eq 12 cannot be applied in a wide range of times. This means that eq 12 can be applied at a shorter range of time as the KL value increases. Figure 3b and c shows that eq 12 can be applied for a longer range of time in comparison to eq 15. Figure 4b and c shows that eqs 12 and 15 can be used in the same range of time. Therefore, it can be concluded that, with increasing the values of KL, the time range of applicability of eq 12 becomes narrow and also closer to that of eq 15. The values of Kls for these two systems were calculated, and they are listed in Table 2. The obtained values of Kls are very close to the original values of Kls by the SRT rate equation. These results prove the accuracy of eqs 12 and 15 once more. In this section, two different sets of experimental desorption kinetic data will be analyzed by the SRT approach. As the first

Solid/Solution Interface Desorption Kinetics

Langmuir, Vol. 24, No. 22, 2008 13017

Figure 4. (a) Numerically generated points (t, q) based on the SRT equation for ka ) 1.00 × 10-5 and kd ) 1.00 × 10-15. (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

system, the desorption kinetic data of vitamin E from silica40 at 40 °C are considered. The experimental kinetic data were fitted to the SRT rate equation (eq 19) by the stochastic numerical simulation method. The input data for simulation are given in Table 1. The values of ka and kd were adjusted until a reasonable fit to the experimental kinetic data was obtained. The obtained values of ka and kd are listed in Table 2. Figure 5a represents the experimental and the simulated (SRT) values of q versus t. The values of θi ln θ - θ(1 + θi) + (θ2/2) as a function of time based on eq 12 were calculated, and they are shown in Figure 5b. Figure 5c shows the linear plot of q versus t1/2 (eq 15). Figure 5b and c shows that in the present system eq 12 can be applied to a slightly larger range of time in comparison to eq 15. This is due to large values of KL for this system. The Kls values were calculated based on eqs 12, 15, and 19, and they are presented in Table 2. There is an agreement between these obtained values, especially between the results of eq 12 and the SRT equation (eq 18). (40) Chu, B. S.; Baharin, B. S.; Cheman, Y. B.; Queck, S. Y. J. Food Eng. 2004, 64, 1.

Figure 5. (a) Desorption kinetics of vitamin E from silica, experimental data (open circles),40 and theoretical data based on the SRT equation (solid line). (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

The other desorption system which has been selected for analysis is the desorption of chromium(VI) from hydrotalcite.41 By applying the experimental kinetic data of this system, the values of ka and kd as adjustable parameters of eq 18 were obtained by stochastic numerical simulation. Figure 6a shows the simulated (SRT) and experimental values of q versus t for this system. Parts b and c of Figure 6 were plotted based on eqs 12 and 15, respectively. It is clear from these figures that eq 12 can be used in a longer range of time in comparison to eq 15. Since the value of KL for this system is small, eq 12 is more appropriate than eq 15 for analysis of initial desorption kinetic data. The values of Kls were calculated by eqs 12, 15, and 18. The obtained values of Kls from the initial time kinetic models (eqs 12 and 15) are very close to the value of Kls obtained by the SRT rate equation. In summary, the experimental data show that the kinetics of desorption can be analyzed by the SRT approach. Both (41) Lazaridis, W. K.; Pandi, T. A.; Matis, K. A. Ind. Eng. Chem. Res. 2004, 43, 2209.

13018 Langmuir, Vol. 24, No. 22, 2008

Azizian and Bashiri

experimental data indicate that the SRT approach is an appropriate method for modeling of desorption kinetics. Two rate equations on the basis of the SRT approach have been derived for initial times of desorption (eqs 12 and 15). By using generated data points (t, q) based on the SRT rate equation, it has been shown that, for KL e 1, eq 12 can be applied to the whole time range of desorption. Also, it has been shown that eq 12 can be applied to a longer range of time in comparison to eq 15. With increasing value of KL, the applicability time range of eq 12 becomes closer to that of eq 15. Both the experimental and generated data approve the accuracy of the derived rate equations at initial times of desorption. However, eq 12 is recommended for analysis of desorption kinetic data at initial times of desorption, because it is more general than eq 15 and at different conditions its results show good agreement with the SRT results.

Nomenclature c cadsorbent ce k K K′ ka kd KD1 KD2 KL Kls Kls′

Figure 6. (a) Desorption kinetics of chromium(VI) from hydrotalcit, experimental data (open circles),41 and theoretical data based on the SRT equation (solid line). (b) Linear plot of eq 12 for initial times of desorption. (c) Linear plot of q versus t1/2 for initial times of desorption.

experimental and theoretical (SRT) data indicate that eqs 12 and 15 are appropriate equations for modeling kinetic data of desorption.

Conclusion The statistical rate theory approach was used for description of desorption kinetics at the solid/solution interface. The

MW q q0 qm qs t T R β β2 µb µb° µs θ θe θi LA8029769

bulk concentration of solute (mol/L) concentration of adsorbent (g/L) bulk equilibrium concentration of solute (mol/L) Boltzmann constant (J/K) constant in eq 13 (min-1) constant in eq 16 (min-1/2) rate constant of adsorption (min-1) rate constant of desorption (mol2/L2 · min) desorption rate constant for the first order model (min-1) desorption rate constant for the second order model (min-1) Langmuir constant (L/mol) constant in eq 8 (L/mol · min) equilibrium rate of adsorption at the solid/solution interface (min-1) molar mass (g/mol) amount of adspecies (mg/g) amount of adspecies at start of desorption (mg/g) maximum value of the amount of adspecies (mg/g) molecular partition function time (min) absolute temperature (K) constant in eq 13 constant in eq 9 (mol/L) constant in eq 2 chemical potential of the adsorbate in the bulk phase standard chemical potential of the adsorbate in the bulk phase chemical potential of the adsorbate in the surface phase fractional surface coverage fractional surface coverage at equilibrium fractional surface coverage at the start of the desorption process