Solution Interface - American

ARTICLE pubs.acs.org/JPCC

Desorption Kinetics at the Solid/Solution Interface: A Theoretical Description by Statistical Rate Theory for Close-to-Equilibrium Systems Hadis Bashiri* Department of Physical Chemistry, Faculty of Chemistry, University of Kashan, Kashan, Iran ABSTRACT: Modeling of desorption kinetics plays an important role for better understanding of this phenomena. In the present work the kinetics of solute desorption at the solid/solution interface has been studied by statistical rate theory (SRT) when the system is close to equilibrium. It is demonstrated that, for desorption, a pseudo-first-order equation can be derived from the SRT equation when the system is close to equilibrium. On the basis of numerically generated points (t; q) by the SRT equation, it is shown that it is not possible to apply the pseudo-first-order equation for modeling of whole kinetic desorption data. The results of numerical analysis are in good agreement with our theoretical derivation of the pseudo-first-order equation for desorption systems close to equilibrium. Finally, the results of the present theoretical study were confirmed by analysis of two experimental systems.

’ INTRODUCTION Sorption affects mass transports in industrial and environmental systems.1-7 Among the various treatment technologies, sorption is one of the most efficient methods, and it was widely used for treatment of industrial wastewater containing heavy metals, colors, and other impurities. The advantages of the sorption process compared to other separation methods are its simplicity in operation and inexpensiveness.1-7 The adsorption and desorption processes are the most common methods for removal of pollutants from waste waters. Sometimes for a contaminant separation process, an adsorption method is used first, and then, to release adsorbed species, a desorption process is needed.8-11 Therefore, desorption kinetics and equilibrium are important in understanding the desorption characteristics from the adsorbent. The kinetics of adsorption has been studied extensively, theoretically and experimentally. Some of the simple empirical models which are applied for the description of adsorption kinetics include the pseudo-first-order,12 pseudo-secondorder,13,14 and modified first-order models,15 the Elovich model,16 multiexponential equation,17 and so on. Theoretical interpretations of some of important adsorption models were provided by Azizian18,19 and Rudzinski and Plazinski20-22 by applying the theory of activated/adsorption desorption (TAAD) and statistical rate theory (SRT). Recently, a review of certain classes of theoretical models was presented by Plazinski et al.23 r 2011 American Chemical Society

There are different models to describe desorption kinetics too. Kinetic equations commonly are used in desorption studies including zero-order,24 pseudo-first-order,25,26 pseudo-secondorder,25,26 and third-order equations,24 the modified Freundlich equation,27 parabolic diffusion,28 two-constant rate,24 and the simple Elovich equation.29 The mentioned models of desorption kinetics were presented experimentally, and there are no theoretical interpretations for them. The pseudo-first-order model was suggested empirically for description of desorption kinetics at the solid/solution interface.25,26 The pseudo-first-order rate equation is25,26 dθ ¼ KD1 ðθe - θÞ dt

ð1Þ

where θ is the fractional surface coverage and is defined as θ = q/ qm. q is the amount of adsorbate at any time, and qm is the maximum value of q. θe is the fractional surface coverage at equilibrium, and KD1 is a constant. The SRT approach has been used for modeling of desorption kinetics at gas/solid30-32 and solid/solution interfaces.33 The analytical solution of the basic equation of SRT for kinetics of desorption at the solid/solution interface led to a complex Received: November 3, 2010 Revised: February 8, 2011 Published: March 10, 2011 5732

dx.doi.org/10.1021/jp110511z | J. Phys. Chem. C 2011, 115, 5732–5739

The Journal of Physical Chemistry C

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expression. Recently, we have derived a simple equation for description of desorption kinetics near the initial times of the process.33 Therefore, desorption kinetics can be modeled based on the SRT approach with simple equations by deriving a desorption rate equation at the close-to-equilibrium condition. This article’s intent is to derive a desorption rate equation on the basis of statistical rate theory at the close-to-equilibrium condition. Solving the statistical rate theory equation close to equilibrium provides a theoretical interpretation for the Lagergren model of desorption kinetics. The theoretical results (derived equations) will provide a definition for the observed rate constant of the pseudo-first-order model (KD1).

By combination of eqs 3-5 and eq 2, one arrives at " ! dθ μ°b θ 0 ¼ K ls exp þ ln qs þ ln c - ln dt 1-θ kT !# θ μ°b - ln qs - - ln c - exp ln 1-θ kT By simplification, the following expression is obtained: dθ cð1 - θÞ 1 θ 0 ¼ Kls KL dt θ KL cð1 - θÞ where KL is the Langmuir constant and is defined as ! μ°b KL ¼ qs exp kT

ð6Þ

ð7Þ

’ THEORY The SRT approach, which is based on quantum mechanics and thermodynamics, has been provided by Ward and coworkers.34,35 The net rate of molecular transport from phase 1 (bulk) to phase 2 (surface) has been calculated by this approach.36 Rudzinski et al. has shown how the SRT approach can further be generalized to describe the kinetics of adsorption at gas/solid37,38 and solid/solution20,21,39 interfaces. Recently, we have used the SRT approach for adsorption kinetics near the initial times of adsorption and close to equilibrium19 and also for description of competitive adsorption at the solid/solution interface.40 The general rate expression to describe the kinetics of adsorption is presented by the following form:20,38 dθ μb - μs μs - μb 0 ¼ K ls exp - exp ð2Þ dt kT kT

Equation 7, which has been presented by Rudzinski and Plazinski, is the basic equation for description of single-solute adsorption on homogeneous solid surfaces.20-22 The SRT has been used by Rudzinski and co-workers for the modeling of desorption kinetics at the gas/solid interface.30-32 Our recent studies have been concerned with the description of the desorption kinetics at the solid/solution interfaces by using the SRT approach.33 It was assumed that, at the start of desorption process, the bulk concentration of solute (c) is equal to zero, but it increases during desorption process as33

where k is the Boltzmann constant, T is absolute temperature, and K0ls is the adsorption/desorption rate at equilibrium. μs and μb are the chemical potentials of the adsorbate on the surface and in the bulk phases, respectively. The first studies to use the SRT approach to describe the sorption rate at the solid/solution interfaces were made by accepting the Langmuir model of one-site-occupancy adsorption on a solid surface.20,22 For the Langmuirian adsorption, the chemical potential of adsorbate on the surface (μs) can be expressed by20

ð10Þ

θ - kT ln qs μs ¼ kT ln 1-θ

ð3Þ

where qs is the molecular partition function of an adsorbed molecule. In solid/solution systems, the chemical potential of adsorbate in the bulk phase (μb) is given by22 μb ¼ μ°b þ kT ln c

0

ð9Þ

where β is a constant and θi is the fractional coverage of adsorbed molecules at the start of the desorption process. By utilizing eq 9, the basic equation of SRT for kinetics of desorption at the solid/ solution interface is expressed as33 dθ βðθi - θÞð1 - θÞ θ 0 ¼ Kls KL dt θ KL βðθi - θÞð1 - θÞ Recently, we have derived the following simple equation for description of desorption kinetics near the initial times of the process based on the above equation.33 θi ln θ - θð1 þ θi Þ þ

θ2 ¼ R0 - k0 t 2

ð11Þ

where R0 and k0 are constants. The aim of this paper is to derive a new and simple equation from eq 10 for desorption kinetics when the system is close to equilibrium. At first eq 10 is rearranged as " # dθ ðθi - θð1 þ θi - θÞÞ θ 0 ¼ Kls KL β - 2 2 dt θ KL β ðθi - θð1 þ θi - θÞÞ

ð4Þ

ð12Þ and for simplification of the above expression, X is defined as

where c is concentration of solute in the bulk phase. The following expression can be assumed for K0ls:21 K ls ¼ Kls ce ð1 - θe Þ

c ¼ βðθi - θÞ

ð8Þ

X 1 þ θi - θ Then eq 12 simplifies to " # dθ ðθi - XθÞ θ 0 ¼ Kls KL β - 2 2 dt θ KL β ðθi - XθÞ

ð5Þ

where, θe is the surface coverage at equilibrium, ce is the equilibrium concentration of solute, and Kls is a constant. As to the parameter Kls, the larger is its value, the faster is adsorption, but in the theoretical calculations it changes only the time scale.

ð13Þ ð14Þ

KL2 in the above equation can be replaced from the Langmuir isotherm. On the basis of eq 9, the equilibrium bulk 5733

dx.doi.org/10.1021/jp110511z |J. Phys. Chem. C 2011, 115, 5732–5739


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concentration (ce) in the Langmuir isotherm is equal to β(θi θe), so " # dθ ðθi - XθÞ θðθi - θe Þ2 ð1 - θe Þ2 0 ¼ Kls KL β ð15Þ dt θ θe 2 ðθi - XθÞ or

"

dθ ðθi - XθÞ θðθi - θe ð1 þ θi - θe ÞÞ2 0 ¼ Kls KL β dt θ θe 2 ðθi - XθÞ

ð19Þ

ð20Þ

and since the system is close to equilibrium, the assumptions of θe þ θ ≈ 2θe and θe/θ ≈ 1 in the last term of eq 20 are acceptable. On the basis of these assumptions eq 20 simplifies to 0

dθ 2K KL βθi ¼ ls 2 ðθe - θÞ ð21Þ dt θe By using the Langmuir isotherm and eq 5, one arrives at dθ 2Kls βθi ¼ ðθe - θÞ ð22Þ dt θe The constant parameters in above equation can be defined as KD1: 2Kls βθi θe

ð23Þ

Substitution of the eq 23 into eq 22 gives eq 1, which is the pseudo-first-order desorption model. In the literature, the pseudo-first-order desorption model has been applied to model the kinetics of desorption over the entire time span of the reaction.24,41,42 As shown in the present derivation, the pseudo-first-order desorption rate equation can be used for modeling of desorption kinetics when the system is close to equilibrium. On the basis of the above derivation it is expected that the application of the pseudo-first-order kinetic model for modeling of desorption kinetics near the initial times of desorption creates large errors. Up to now the integration of the pseudo-first-order kinetic rate equation has been done by the boundary condition θ(t = 0)

ð24Þ

Equation 24 can be simplified as lnðθe - θÞ ¼ R - KD1 t

Since the system is close to the equilibrium, it can be assumed that Xe = X. By this assumption eq 18 converts to " # dθ θe 2 θi 2 - 2Xθθe 2 θi - θ2 θi 2 þ 2Xθe θ2 θi 0 ¼ Kls KL β dt θθe 2 ðθi - XθÞ

KD1

lnðθe - θÞ ¼ lnðθe - θ1 Þ - KD1 t1 - KD1 t

#

ð16Þ On the basis of eq 13, Xe can be defined as Xe 1 þ θi - θe. Therefore, eq 16 can be written as " # dθ ðθi - XθÞ θðθi - θe Xe Þ2 0 ¼ Kls KL β - 2 ð17Þ dt θ θe ðθi - XθÞ By rearrangement of eq 17, one arrives at " # dθ θe 2 ðθi - XθÞ2 - θ2 ðθi - Xe θe Þ2 0 ¼ Kls KL β ð18Þ dt θθe 2 ðθi - XθÞ

By simplification of the above equation, one arrives at dθ θi θi ðθe þ θÞ - 2Xθθe 0 ¼ Kls KL β 2 ðθe - θÞ dt θðθi - XθÞ θe

= 0.24,41,42 On the basis of the present derivation, the integration of the mentioned kinetic model should be done with the boundary condition θ(t1) = θ1, where t1 is the time after which the pseudo-first-order kinetic model can be used and t1 6¼ 0. Therefore, integration of eq 1 with the boundary condition θ(t1) = θ1 yields

ðfor t g t1 Þ

ð25Þ

where R is a constant and is equal to R ¼ lnðθe - θ1 Þ - KD1 t1 ð26Þ As shown in eq 25, it is expected that the plot of ln(θe - θ) is a linear function of time (for t g t1). The intercept of this plot is R, and the tangent is -KD1. It has been concluded that the pseudo-first-order rate equation of “desorption” can be derived theoretically based on the SRT equation when the system is close to equilibrium. Previously, the pseudo-first-order equation for “adsorption” was derived based on the SRT approach when the system is close to equilibrium.20 It is interesting to note that the Lagergren kinetic model can be derived for both desorption and adsorption systems, when these systems are close to equilibrium. In summary, it can be concluded theoretically that the pseudofirst-order kinetic model can be applied for modeling of kinetic data in adsorption or desorption systems when the system is close to equilibrium. On the basis of the Azizian derivation of the Lagergren kinetic equation, this model can be applied for modeling of adsorption kinetic data when the initial concentration of solute is high compared to βθ.18 In the next section by using generated and experimental data the accuracy of the present derivations will be discussed.

’ RESULTS AND DISCUSSION In this section, we are going to analyze the applicability of the derived equation (eq 25). For this purpose three sets of hypothetical kinetic data points (q; t) were generated based on the SRT rate equation for desorption at a solid/solution interface (eq 10). The analytical solution of eq 10 led to a complex expression. One powerful technique for numerical simulations is stochastic simulation. For this purpose we applied the CKS package developed by Houle and Hinsberg.43,44 In this method the reaction mechanism is considered as a collection of several steps:43 nN þ mM þ ... f products ð27Þ The input data for simulation are the steps and their rate constants, ki. The rate of the ith step, Ri, is taken to be proportional to the probability, Pi, of its occurring in a particular time interval. Pi µ Ri ¼ ki ½Nn ½Mm

ð28Þ

The time step Δt between occurrences of any of the reaction steps is the mean time for a system obeying Poisson statistics: Δt ¼

-ln F

∑R i

ð29Þ

where F is a random number between 0 and 1. The simulation is propagated by repetitively selecting at random among the 5734



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Table 1. Constants Which Have Been Used for Desorption Kinetic Modeling system

qi (mg/g)

qm (mg/g)

Mw (g/mol)

cadsorbent (g/L)

SRT (generated data)

60.00

60.00

60.00

10.00

vitamin E/silica

53.86 10-1

36.21 10-3

43.07 101

20.00

Cd(II)/A. caviae

37.64

15.53 101

11.20 101

10.00 10-1

Table 2. Values of ka and kd Used for Different Systems and the Kls Values Obtained for Different Systems Based on Different Kinetic Models Kls system generated data 1 (by SRT)

ka (min-1) 1.00 10

-5 -5

kd (mol2/L2 3 min) 1.00 10

-5

1.00 10

-4

SRT

eq 25

0.61 10

-1

0.62 10-1

generated data 2 (by SRT)

1.00 10

0.17

0.18

generated data 3 (by SRT)

1.00 10-5

1.00 10-6

0.22 10-1

0.23 10-1

vitamin E/silica

1.00 10-3

1.00 10-13

0.26 105

0.33 105

0.13 10

0.11 102

Cd(II)/A. caviae

2.80 10

-3

-9

2.15 10

2

probability-weighted steps in the mechanism and updating the reactant and product populations according to the stoichiometry of the selected step, system state variables, and reaction rates.43 The result is a set of concentration versus time curves that may be compared directly to experiment. This stochastic numerical simulation method has been used to simulate several chemical kinetics reactions44-47 including surface processes.43,48 Recently, we utilized this method for numerical solution of some adsorption and desorption kinetic equations, successfully.19,35,39 For simplification of eq 10 the adsorption and desorption rate constants (ka and kd) are defined as 0

ka ¼ Kls KL

ð30Þ

0

Kls KL Combination of eq 10 and the above equations gives

Figure 1. (a) Numerically generated data points (t; q) based on the SRT equation for ka = 1.00 10-5 min-1 and kd = 1.00 10-5 mol2/ L2 3 min. (b) Relative errors of calculated θ values based on eq 25 when the kinetic data in the entire time span of desorption were applied. (c) Linear plot of eq 25 when the system is close to equilibrium.

kd ¼

ð31Þ

dθ βðθi - θÞð1 - θÞ θ ¼ ka - kd dt θ βðθi - θÞð1 - θÞ

ð32Þ

The generated data were obtained by considering certain values of qi (the amount of adsorbate on the surface at the start of desorption), qm, ka, and kd (Tables 1 and 2).33 Three sets of desorption kinetic data points (q; t) with similar initial conditions (Tables 1 and 2) but different kd values (Table 2) were generated. The different values of kd were used to make different values of the Langmuir equilibrium constant (KL). In the first system, ka is equal to kd (ka = 1.00 10-5 min-1, kd = 1.00 10-5 mol2/L2 3 min). The generated data (q; t) based on the SRT equation (eq 10) are shown in Figure 1a. In the last section, it is shown theoretically that the pseudo-first-order rate equation can be derived for desorption kinetic data when the system is close to equilibrium. Now we are going to show numerically that we cannot apply the pseudo-first-order equation to model the kinetic data over the entire time span of desorption. Up to now the pseudo-first-order kinetic model has been used for modeling the kinetic data over the entire time span of desorption.41,42 With utilization of generated data, values of ln(θe - θ) as a function of time were calculated (see eq 24) and plotted in the 5735



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Table 3. Comparison between the Obtained Values of θe Based on the SRT Rate Equation and/or Experimental Data with eq 25 for Different Systems θe system generated data 1 (by SRT) generated data 2 (by SRT) generated data 3 (by SRT) vitamin E/silica Cd(II)/A. caviae

obtained from SRT

obtained from eq 25

-1

0.16 10-1

-2

0.60 10-2

-1

0.45 10-1

-1

0.23 10-1

-1

1.54 10-1

0.19 10

0.63 10 0.45 10 0.28 10

1.53 10

Figure 3. (a) Numerically generated points (t; q) based on the SRT equation for ka = 1.00 10-5 min-1 and kd = 1.00 10-6 mol2/ L2 3 min. (b) Relative errors of calculated θ values based on eq 25 when the kinetic data in the entire time span of desorption were applied. (c) Linear plot of eq 25 when the system is close to equilibrium.

The relative error (RE) which is used to evaluate capability of kinetic equations for modeling of kinetic data defined as19 RE ¼

Figure 2. (a) Numerically generated data points (t; q) based on the SRT equation for ka = 1.00 10-5 min-1 and kd = 1.00 10-4 mol2/ L2 3 min. (b) Relative errors of calculated θ values based on eq 25 when the kinetic data in the entire time span of desorption were applied. (c) Linear plot of eq 25 when the system is close to equilibrium.

entire time span of desorption (plot not shown here). In this case the boundary condition is θ(t = 0) = 0. Therefore, the intercept and slope of this plot are equal to ln θe and KD1, respectively. Now on the basis of obtained values of θe and KD1, the values of θ were calculated as a function of time.

θcalcd - θexptl θexptl

ð33Þ

where θexptl is the experimental (or in the present case, the generated data by the SRT rate equation) value of θ and θcalcd is the fractional surface coverage calculated by the kinetic model (in the present case, the pseudo-first-order model). The calculated relative errors as a function of time are shown in Figure 1b. As shown in this figure, near the initial times of desorption the value of relative error is large and decreases as time increases. This plot shows that we cannot use the pseudo-first-order model for the initial times of desorption. This numerical analysis is in perfect agreement with our theoretical derivation of the pseudo-firstorder equation by the SRT equation. For close-to-equilibrium kinetic data, the values of ln(θe - θ) versus t are represented in Figure 1c. The value of θe was adjusted until the intercept of the linear plot became equal to R (eq 25). 5736



Figure 4. (a) Desorption kinetics of vitamin E from silica: experimental data (open circles) (ref 49) and theoretical data based on the SRT equation (solid line). (b) Relative errors of calculated θ values based on eq 25 when the kinetic data in the entire time span of desorption were applied. (c) Linear plot of eq 25 when the system is close to equilibrium.

The adjustable value of θe (obtained from the pseudo-first-order equation) is reported in Table 3. A comparison between this value of θe and the obtained value from SRT is shown in Table 3. As shown in this table, the obtained value of θe is in good agreement with the original one (obtained from SRT). By utilizing the slope of Figure 1c (KD1 constant), the value of Kls was calculated by eq 23 and is reported in Table 2. The Kls value was obtained from the SRT rate equation by eqs 5, 30, and 32 and is reported in Table 2. Agreement between the obtained Kls values from the SRT rate equation and from eq 23 confirms the applicability of the pseudo-first-order model for close-to-equilibrium desorption kinetic data. Two other sets of kinetic data points were generated by ka < kd (ka = 1.00 10-5 min-1, kd = 1.00 10-4 mol2/L2 3 min) and ka > kd (ka = 1.00 10-5 min-1, kd = 1.00 10-6 mol2/L2 3 min). The generated data points (q; t) of these two systems are shown in Figures 2a and 3a. The values of θe and KD1 were obtained by plotting of ln(θe - θ) versus time in the entire time span of desorption (graph not presented here). By using these data and

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Figure 5. (a) Desorption kinetics of cadmium(II) from A. caviae: experimental data (open circles) (ref 50) and theoretical data based on the SRT equation (solid line). (b) Relative errors of calculated θ values based on eq 25 when the kinetic data in the entire time span of desorption were applied. (c) Linear plot of eq 25 when the system is close to equilibrium.

the Lagergren model, the values of θ as a function of time were calculated and then the relative errors were obtained. The relative errors of these values of θ were calculated by eq 33 and are shown in Figures 2b and 3b. Again these figures show that application of the pseudo-first-order kinetic model for whole kinetic data creates large errors near the initial time of desorption. The values of ln(θe - θ) versus t for close-to-equilibrium desorption kinetic data are plotted in Figures 2c and 3c. The values of θe were obtained as adjustable parameters from the pseudo-first-order equation and are listed in Table 3. As shown in this table, the obtained values of θe are in good agreement with the original ones (obtained from SRT). Also the Kls values calculated from these results are listed in Table 2. The comparison between the obtained Kls values and the original ones (calculated by the SRT rate equation) proves the accuracy of the above derivation again. In this section two different sets of experimental data have been selected from the literatures49,50 to be analyzed by the SRT 5737


The Journal of Physical Chemistry C approach. The first experimental system that was considered is from the paper of Chu et al.,49 which reports a batch desorption study of vitamin E from silica. The experimental kinetic data at 40 °C were fitted to the SRT rate equation (eq 32) by the stochastic numerical simulation method. The input data for simulation are written 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 reported in Table 2. The experimental and simulated values of q as a function of time are represented in Figure 4a. The values of ln(θe - θ) as a function of time were plotted in the entire time span of desorption, and then values of KD1 and θe were obtained. Then by utilizing these results and eq 24, the values of θ were calculated, and the relative errors of these data are plotted in Figure 4b. As expected, using the Lagergren model in the entire time span of desorption for this system large errors appear near the initial times. By using of the pseudo-first-order model for desorption kinetic data close to equilibrium (Figure 4c), the Kls and θe values were obtained and are listed in Tables 2 and 3. The obtained values of Kls and θe are close to the original ones (Kls and θe obtained from the SRT rate equation). The obtained results of these experimental data prove the accuracy of our derivation for the desorption pseudo-first-order model. The next experimental desorption system that has been selected to be analyzed is desorption of cadmium(II) from Aeromonas caviae.50 The results of fitting of experimental data by the SRT desorption rate equation are shown in Figure 5a. The values of ka and kd are obtained as adjustable parameters and are listed in Table 2. Application of the Lagergren equation in the entire time span of desorption for this system causes large errors near the initial times, too (Figure 5b). Utilizing the pseudo-firstorder model for close-to-equilibrium kinetic data of this system (Figure 5c) gives acceptable values of Kls and θe (Tables 2 and 3). In summary, both experimental and theoretical (SRT) data are in perfect agreement with our theoretical derivation and also show that the pseudo-first-order model can be applied for modeling of desorption kinetic data when the system is close to equilibrium.

’ CONCLUSION The desorption kinetics at solid/solution interfaces has been studied based on the SRT approach at close-to -equilibrium conditions. By considering some assumptions on the SRT rate equation, a pseudo-first-order kinetic model was derived for desorption at the solid/solution interface. It has been shown theoretically that the Lagergren model can be derived from the SRT equation when the desorption system is close to equilibrium. Three different sets of numerical desorption kinetic data (q; t) were generated by using the SRT rate equation. By utilizing these generated data, it was shown that we cannot use the pseudo-firstorder model in the entire time span of desorption. On the basis of two experimental kinetic data, it was indicated that modeling of kinetic data by the Lagergren model in the entire time span of desorption creates large errors near the initial times. Both experimental and generated data were in good agreement with our derivation. For modeling of desorption kinetics based on the SRT approach, but by simple equations, one has to use eq 11 for

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initial times of desorption and eq 25 for the close-to-equilibrium condition.

’ AUTHOR INFORMATION Corresponding Author

*Fax: þ98 361 5552930. E-mail: [email protected] and h. [email protected].

’ ACKNOWLEDGMENT The author gratefully acknowledges Professor S. Azizian from Bu-Ali Sina University for helpful discussions. ’ NOMENCLATURE c bulk concentration of solute (mol/L) cadsorbent concentration of adsorbent (g/L) bulk equilibrium concentration of solute (mol/L) ce k Boltzmann constant (J/K) rate constant of adsorption (min-1) ka rate constant of desorption (mol2/L2 3 min) kd 0 constant in eq 11 k desorption rate constant for the pseudo-first-order KD1 model (min-1) Langmuir constant (L/mol) KL constant in eq 8 (L/mol 3 min) Kls the equilibrium rate of adsorption at the solid/solution K0ls interface (min-1) molar mass (g/mol) MW q the amount of adspecies per unit mass of adsorbent (mg/g) the amount of adspecies per unit mass of adsorbent at qi the start of desorption (mg/g) maximum value of the amount of adspecies per unit qm mass of adsorbent (mg/g) molecular partition function qs t time (min) T absolute temperature (K) R constant in eq 25 constant in eq 11 R0 β constant in eq 9 (mol/L) chemical potential of the adsorbate in the bulk phase μb standard chemical potential of the adsorbate in the μb° bulk phase chemical potential of the adsorbate on the μs surface phase θ fractional surface coverage fractional surface coverage at equilibrium θe fractional surface coverage at the start of the desorption θi process ’ REFERENCES (1) Dzombak, D. A.; Morel, F. M. M. J. Colloid Interface Sci. 1986, 112, 588. (2) Yin, Y.; Allen, H. E.; Huang, C. P. Environ. Sci. Technol. 1997, 31, 496. (3) Ho, Y. S.; McKay, G. Trans. Inst. Chem. Eng. 1998, 76, 332. (4) Ho, Y. S.; McKay, G. Trans. Inst. Chem. Eng. 1999, 77, 165. (5) Prasad, M.; Saxena, S.; Amritphale, S. S.; Chandra, N. Ind. Eng. Chem. Res. 2000, 239, 3034. (6) Chenug, C. W.; Porter, J. F.; McKay, G. Sep. Purif. Technol. 2000, 19, 55. 5738



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