Langmuir 2008, 24, 11669-11676
11669
Adsorption Kinetics at the Solid/Solution Interface: Statistical Rate Theory at Initial Times of Adsorption and Close to Equilibrium Saeid Azizian* and Hadis Bashiri Department of Physical Chemistry, Faculty of Chemistry, Bu-Ali Sina UniVersity, Hamedan 65174, Iran ReceiVed July 17, 2008. ReVised Manuscript ReceiVed August 9, 2008 The kinetics of solute adsorption at the solid/solution interface has been studied by statistical rate theory (SRT) at two limiting conditions, one at initial times of adsorption and the other close to equilibrium. A new kinetic equation has been derived for initial times of adsorption on the basis of SRT. For the first time a theoretical interpretation based on SRT has been provided for the modified pseudo-first-order (MPFO) kinetic equation which was proposed empirically by Yang and Al-Duri. It has been shown that the MPFO kinetic 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 has been shown that we can apply the new equation for initial times of adsorption in a larger time range in comparison to the previous q vs t linear equation. Also by numerical analysis of the generated kinetic data points, it is shown that application of the MPFO equation for modeling of whole kinetic data causes a large error for the data at initial times of adsorption. The results of numerical analysis are in perfect agreement with our theoretical derivation of the MPFO kinetic equation from the SRT equation. Finally, the results of the present theoretical study were confirmed by analysis of an experimental system.
Introduction Adsorption is one of the important methods for gas separation and wastewater purification. The design of industrial adsorption equipment requires to theoretical and experimental study of this process. Both kinetics and equilibrium are important in adsorption studies. The equilibrium adsorption (thermodynamics) has been studied extensively, and various isotherm equations have been reported for this.1-4 Recently, we proposed a new isotherm equation which accounts for two different states of an adsorbate.5 There are different and simple models for adsorption kinetics at the solid/solution interface. These models include the pseudofirst-order model,6 pseudo-second-order model,7,8 and modified pseudo-first-order (MPFO) model.9 All of these kinetic equations were proposed empirically. Recently, some attempts were made to find a theoretical basis for these equations. In 2004, on the basis of the theory of activated adsorption/desorption (TAAD), Azizian derived the pseudo-first-order and the pseudo-secondorder kinetic models.10 Also he showed theoretically10 and experimentally11 that the pseudo-second-order rate constant is a complex function of the initial concentration of the solute. Then Rudzinski and Plazinski tried to find a theoretical basis for the pseudo-first-order and the pseudo-second-order kinetic models on the basis of statistical rate theory (SRT).12,13 In 2006, Azizian developed the pseudo-second-order model for heterogeneous * To whom correspondence should be addressed. Fax: +98-811-8257404. E-mail:
[email protected] and
[email protected]. (1) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (2) Freundlich, H. Kapillarchemie; Akademische Verlagsgesellschaft: Leipzig, Germany, 1992. (3) Sips, R. J. Chem. Phys. 1948, 16, 490. (4) Quinones, I.; Guiochon, G. J. Colloid Interface Sci. 1996, 57, 183. (5) Azizian, S.; Volkov, A. G. Chem. Phys. Lett. 2008, 454, 409. (6) Lagergren, S. Kungliga SVenska Vetenskapsakademiens, Handlingar 1898, 24, 1. (7) Ho, Y. S.; Mckay, G. Water Res. 2000, 34, 735. (8) Blanchard, G.; Maunaye, M.; Martin, G. Water Res. 1984, 18, 1501. (9) Yang, X.; Al-Duri, B. J. Colloid Interface Sci. 2005, 287, 25. (10) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (11) Azizian, S.; Yahyaei, B. J. Colloid Interface Sci. 2006, 299, 112. (12) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514. (13) Rudzinski, W.; Plazinski, W. Appl. Surf. Sci. 2007, 253, 5827.
adsorbents.14 Most recently, Azizian et al. reported an approximate method to obtain the rate constants of Langmuir-Freundlich kinetics.15,16 Yang and Al-Duri modified empirically the pseudo-first-order model for the kinetics of adsorption at the solid/solution interface.9 The MPFO rate equation is9
θe dθ ) KM (θe - θ) dt θ
(1)
where θ is the fractional surface coverage and is defined as θ ) q/qm. q is the amount of adsorbate, and qm is the maximum value of q. θe is the fractional surface coverage at equilibrium, and KM is a constant. Recently, Rudzinski and Plazinski17 tried to find a theoretical background for the MPFO equation by comparison of the SRT and MPFO data. Up to now the MPFO equation was not derived theoretically. The purpose of this work is to solve the adsorption rate equation on the basis of SRT at two limits, one at initial times of adsorption and the other one close to equilibrium. For the first time, a theoretical interpretation is derived for the MPFO rate equation. The advantage of the present derivation is to show the condition where the MPFO rate equation can be used, and also a definition is provided for KM.
Theory The SRT approach, which is based on quantum mechanics and thermodynamics, has been provided by Ward.18,19 In this approach, the rate of adsorption at the interface is predicted.18,19 Rudzinski et al. applied the SRT approach for adsorption at gas/solid20,21 and solid/solution12,22 interfaces. Also recently we (14) Azizian, S. J. Colloid Intefrace Sci. 2006, 302, 76. (15) Azizian, S.; Haerifar, M.; Basiri-parsa, J. Chemosphere 2007, 68, 2040. (16) Azizian, S.; Haerifar, M.; Bashiri, H. Chem. Eng. J. [Online early access]. DOI: 10.1016/j.cej.2008.05.024. (17) Rudzinski, W.; Plazinski, W. Langmuir 2008, 24, 5393. (18) Ward, C. A. J. Chem. Phys. 1977, 67, 229. (19) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (20) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (21) Panczyk, T.; Rudzinski, W. J. Phys. Chem. B 2004, 108, 2898. (22) Rudzinski, W.; Plazinski, W. J. Phys. Chem. C 2007, 111, 15100.
10.1021/la802288p CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
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Azizian and Bashiri
developed the SRT approach for competitive adsorption at the solid/solution interface.23 The rate of adsorption at the solid/solution interface and based on the SRT approach is represented by the following expression:12,18
[ (
)
(
µb - µs µs - µb dθ - exp ) Kls ′ exp dt kT kT
)]
θ - kT ln qs 1-θ
(4)
where qs is the molecular partition function of the adspecies. By substitution of eqs 3 and 4 into eq 2
[
dθ c(1 - θ) θ 1 ) Kls ′ KL dt θ KL c(1 - θ) where KL is defined as
]
( )
KL ) qs exp
µb ° kT
(5)
(6)
Also Kls′ can be defined as22
Kls ′ ) Klsce(1 - θe)
tf0
dθ 1 ) Kls ′ KLc0 dt θ
(10)
[
]
(11)
KL2c02 - θ2 dθ ) Kls ′ dt KLc0θ
Integration of eq 11 with the boundary condition θ(t)0) ) 0 gives
ln(KL2c02 - θ2) ) ln(KL2c02) -
2Kls ′ t KLc0
(12)
This new equation shows that, at initial times of adsorption, the plot of ln(KL2c02 - θ2) vs t should be linear. The slope of this line is -2Kls′/KLc0, and its intercept is ln(KL2c02). Both the slope and intercept of this plot are a function of the initial concentration of the solute. Equation 12 is a new equation that shows the relation between q and time at the initial times of adsorption. If θ , KLc0, then eq 10 will convert to eq 8, which was derived previously by Rudzinski and Plazinski.24 Therefore, it can be concluded that eq 12 is more general than eq 9 for the initial times of adsorption. In the section on the analysis of the experimental data we will discuss more about the advantages of eq 12. (b) Adsorption Kinetics Close to Equilibrium. The bulk concentration of the solute decreases with increasing surface coverage (θ) as
c ) c0 - βθ
(13)
where β is a constant. By inserting eq 13 into eq 5, one arrives at
[
]
(c0 - βθ)(1 - θ) dθ θ ) Kls ′ KL - 2 dt θ KL (c0 - βθ)(1 - θ) (14) Equation 14 can be rearranged to
[
(c0 - θ(β + c0 - βθ)) dθ ) Kls ′ KL dt θ θ KL2(c0 - θ(β + c0 - βθ))
]
(15)
For simplification of the above expression, A is defined as
(8)
Integration with the boundary condition θ(t)0) ) 0 gives
θ ) √2Kls ′ KLc0t
or
(7)
where ce is the equilibrium concentration of the solute. Next, the SRT adsorption rate equation (eq 5) will be analyzed at two limiting conditions: (a) initial times and (b) close to equilibrium. (a) Adsorption Kinetics at Initial Times. Most recently, Rudzinski and Plazinski solved the SRT equation (eq 5) at the very short initial times of adsorption24 for the systems where the surface reaction is the rate-controlling step and the diffusion in the subsurface region plays no role in the adsorption kinetics. For this purpose they considered the following assumptions on eq 5: (i) The bulk concentration of the solute is equal to its initial value; i.e., c = c0, where c0 is the initial concentration of the solute. (ii) The θ value can be ignored in the 1 - θ term. (iii) The desorption rate (last term in eq 5) is ignorable. Therefore, in the limit t f 0, eq 5 simplifies to
lim
]
(3)
where c is the concentration of the solute in the solution phase. If the equilibrium adsorption data follow the Langmuir isotherm, then µs for the adsorbate will be12
µs ) kT ln
[
c0 θ dθ ) Kls ′ KL dt θ KLc0
(2)
where Kls′ is the rate of adsorption at the solid/solution interface and at equilibrium, µs and µb are the chemical potentials of the adsorbate on the surface and in the bulk phase, respectively, k is the Boltzmann constant, and T is the absolute temperature. In solid/solution systems, µb is calculated by12
µb ) µb ° + kT ln c
step on the basis of SRT. For this purpose we considered two assumptions including (i) and (ii) above. In other words we did not ignore the desorption term in eq 5. In this case eq 5 converts to
A ≡ β + c0 - βθ
(16)
Then eq 15 simplifies to
(9)
Thus, at the very short initial times of adsorption, θ is a linear function of t and the tangent of its linear plot is (2Kls′KLc0). A similar θ vs t linear dependence is also predicted for the interparticle-diffusion-controlled adsorption.25 In this section we derive a new kinetic equation for the initial times of adsorption when the surface reaction is the controlling (23) Azizian, S.; Bashiri, H.; Iloukhani, H. J. Phys. Chem. C 2008, 112, 10251. (24) Rudzinski, W.; Plazinski, W. Langmuir 2008, 24, 6738. (25) Suzuki, M. Adsorption Engineering; Kodansha: Tokyo, 1990.
[
(c0 - Aθ) dθ θ ) Kls ′ KL - 2 dt θ KL (c0 - Aθ)
]
(17)
KL2 in the last term of the above equation can be replaced from the Langmuir isotherm:
[
(c0 - Aθ) θ(1 - θe)2ce2 dθ ) Kls ′ KL - 2 dt θ θ (c - Aθ) e
0
]
(18)
On the basis of eq 13, ce is equal to c0 - βθe, and therefore, ce in eq 18 can be replaced by c0 - βθe, so
Adsorption Kinetics at the Solid/Solution Interface
[
(c0 - Aθ) θ(c0 - θe(β + c0 - βθe))2 dθ ) Kls ′ KL dt θ θ 2(c - Aθ) e
0
]
(19)
Langmuir, Vol. 24, No. 20, 2008 11671
ln(θe - θ) +
[
e
0
]
(20)
By rearrangement of eq 20
dθ dt
[
θe2(c0 - Aθ)2 - θ2(c0 - Aeθe)2 ) Kls ′ KL θθe2(c0 - Aθ)
]
Since the system is close to equilibrium, one can assume Ae = A; by this assumption, eq 21 simplifies to
[
(θe - θ) c0(θe + θ) - 2Aθθe dθ ) Kls ′ KLc0 dt θ θe2(c0 - Aθ)
]
(22)
Since the system is close to equilibrium, the assumption of θe + θ ≈ 2θe is acceptable and eq 22 can be written as
[
(θe - θ) 2c0θe - 2Aθθe dθ ) Kls ′ KLc0 dt θ θe2(c0 - Aθ)
]
(23)
2Klsc0 θe
Results and Discussion In the first step, the adsorption and desorption rate constants are defined as
kd )
(26)
Kls ′ KL
(32)
dθ c(1 - θ) θ ) ka - kd dt θ c(1 - θ)
(33)
In this section, we are going to analyze the applicability of the derived equations (eqs 12 and 30). For this purpose, at first we generated numerically the points (t, q) by using the SRT equation (eq 33) and considering certain values of c0, ka, and kd. The generation of kinetic data points using eq 33 was performed by stochastic numerical simulation. The CKS package developed by Houle and Hinsberg26 was used for stochastic numerical simulation of kinetic data points. Recently, we applied this method for simulation of the adsorption kinetics at the solid/solution interface.16,23 Three different sets of kinetic data points were generated with similar initial conditions (Tables 1 and 2), but different kd values Table 1. Some of the Used Input Data (Constants) for Numerical Generation of Kinetic Data for Three Different Systems c0 (mol/ L) 1.00 × 10
-4
qm (mg/g)
Mw (g/mol)
cadsorbent (g/L)
42.00
60.00
0.10
Table 2. Values of ka and kd Used for Generation of Three Different Systems and Kls Values Obtained for Different Systems Based on Different Kinetic Models
θ1 θ ) ln(θe - θ1) + + KMt1 - KMt (27) θe θe
where t1 is the time after which the MPFO rate equation can be applied, and t1 * 0. Equation 27 can be written in a simplified form as
(31)
By substitution of eqs 31 and 32 into eq 5, one arrives at
(25)
Combination of eqs 25 and 26 gives eq 1, which is the MPFO rate equation. Up to now the MPFO equation has been applied for modeling of adsorption kinetic data in the whole range of adsorption times, but the present derivation shows that the MPFO rate equation can be used for modeling of adsorption kinetics when the system is close to equilibrium. On the basis of the present derivation, it is expected that the application of the MPFO rate equation for modeling of the kinetic data at initial times of adsorption is not correct. Integration of eq 1 has been done with the boundary condition θ(t)0) ) 0 previously,9,17 but on the basis of the above derivation, the integration should be done with the boundary condition θ(t1) ) θ1. Therefore, integration of eq 1 with the boundary condition θ(t1) ) θ1 yields
ln(θe - θ) +
(30)
where R′ is equal to ln(qe - q1) + q1/qe + KMt1. Therefore, the linear diagram of ln(qe - q) + q/qe vs t (for t g t1) can be obtained with changing qe as an adjustable parameter. In summary, it has been concluded that, by accepting the Langmuir isotherm for equilibrium data, the MPFO equation can be derived theoretically on the basis of the SRT equation when the system is close to equilibrium. It is interesting to note that by replacing θ in the denominator of eq 21 or eq 24 by θe the pseudo-first-order equation is obtained.
(24)
The constant parameters of eq 25 can be defined as KM:
KM )
q ) R - KMt qe
ka ) Kls ′ KL
By using eq 7 and also the Langmuir isotherm, one arrives at
dθ 2Klsc0 θe ) (θ - θ) dt θe θ e
(29)
Thus, it is expected that the plot of ln(θe - θ) + θ/θe is a linear function of time (for t g t1). The tangent and intercept of this plot are -KM and R, respectively. By using the definition of θ ) q/qm, eq 28 converts to
or
dθ 2Kls ′ KLc0 (θe - θ) ) dt θe θ
θ1 + KMt1 θe
R ) ln(θe - θ1) +
ln(qe - q) + (21)
(28)
where R is a constant and is equal to
On the basis of eq 16, Ae can be defined as β + c0 - βθe, and therefore, eq 19 converts to
(c0 - Aθ) θ(c0 - Aeθe)2 dθ ) Kls ′ KL - 2 dt θ θ (c - Aθ)
θ ) R - KMt θe
Kls
system 1 2 3
ka (min-1)
kd [mol2/ (L2 · min)]
qe (mg/g)
SRT
initial times (eq 12)
MPFO model (eq 30)
1.00 × 10-5 1.00 × 10-5 4.14 × 10-3 1.45 × 103 1.47 × 103 1.55 × 103 1.00 × 10-5 1.00 × 10-10 1.26 4.76 4.46 4.32 1.00 × 10-5 1.00 × 10-4 1.33 × 10-3 4.51 × 103 4.60 × 103 4.02 × 103
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Figure 1. (a) Numerically generated points (t, q) based on the SRT equation, ka ) kd ) 1.00 × 10-5. (b) Linear plot of the new initial rate equation for initial times of adsorption. (c) Linear plot of q vs t for initial times of adsorption. (d) Relative error of calculated q values based on the MPFO equation when the kinetic data in the whole range of time were applied. (e) Linear plot of the MPFO equation when the system is close to equilibrium. Table 3. Obtained Values of qe and KM from the MPFO Kinetic Equation When Two Different Sets of Data Were Employed: (i) All Kinetic Data and (ii) Close to Equilibrium Data (i) “all” data 1 2 3
(ii) “close to equilibrium” data -1
qe (mg/g)
KM (min )
qe (mg/g)
KM (min-1)
4.32 × 10-3 1.28 1.35 × 10-3
2.03 × 103 2.43 × 10-2 2.12 × 104
4.20 × 10-3 1.26 1.33 × 10-3
3.10 × 103 2.87 × 10-2 2.54 × 104
(Table 2). The different values of kd led to different values of KL (Langmuir constant). In other words, we generated three (26) Houle, F. A.; Hinsberg, W. D. Surf. Sci. 1995, 338, 329.
systems with different affinities between the adsorbate and adsorbent. The values of qe, which were obtained by simulation, are listed in Table 2. The generated data by eq 33 were used to analyze the applicability of limiting rate equations (eqs 12 and 30). In the first system, ka is equal to kd (ka ) kd ) 1 × 10-5 [mol2/(L2 · min)]). The generated data based on the SRT rate equation are shown in Figure 1a. With utilization of these data, the values of ln(KL2c02 - θ2) as a function of time were calculated (see eq 12) and are represented in Figure 1b. As shown in Figure 1b, the diagram is linear with a high correlation coefficient (R2 ) 0.9998). Figure 1c shows the plot of q vs t based on the
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Figure 2. Application of numerically generated points (t, q) based on the SRT equation for ka ) 1.00 × 10-5 and kd ) 1.00 × 10-10 in different cases: (a) linear plot of the new initial rate equation for initial times of adsorption, (b) linear plot of q vs t for initial times of adsorption, (c) relative error of calculated q values based on the MPFO equation when the kinetic data in the whole range of time were applied, (d) linear plot of the MPFO equation when the system is close to equilibrium.
previous rate equation for initial times of adsorption (eq 9). To have a line with a high correlation coefficient, we can apply the data up to 4.4 × 10-5 and 1.0 × 10-3 min for eqs 9 and 12, respectively. This means that eq 12 can be used in a longer time range in comparison to eq 9. The value of Kls can be obtained by initial kinetic data and on the basis of eqs 7, 12, and 31 and the slope of Figure 1b. The calculated values of Kls based on initial kinetic data are listed in Table 2. Comparison between the obtained Kls values from the SRT rate equation and from the initial rate model (eq 12) shows that this new model is suitable for analysis of kinetic data at initial times of adsorption. In the Theory section, we showed theoretically that the MPFO rate equation can be derived from the SRT equation when the system is close to equilibrium. Now it will be shown numerically that we cannot apply the MPFO equation for adsorption kinetic data in the whole range of time. In the literature the MPFO equation has been used for the whole range of time.9,17 On the basis of the previous approach of the MPFO equation, ln(qe - q) + q/qe vs t was plotted in the whole range of time (plot not shown here). In this case the boundary condition is q(t)0) ) 0. Therefore, the intercept of this linear plot is equal to ln qe. The value of qe was considered as an adjustable parameter. The slope of this graph is KM, so the obtained values of qe and KM are listed in Table 3. Comparison of the obtained qe values from the MPFO equation (Table 3) and the original ones (Table 2) shows that they are different. As will be shown, this is due to the application of kinetic data in the whole range of time for modeling with the MPFO equation.
Now on the basis of the data of Table 3 and the MPFO equation, q as a function of time was calculated. To evaluate the capability of kinetic equations for modeling of adsorption kinetic data, one can use the relative error (RE) defined as
RE )
qcalcd - qexptl qexptl
(34)
where qcalcd is the amount of the adspecies calculated by the kinetic model (in the present case, the MPFO model) and qexptl is the experimental (or in the present case, the generated data by the SRT rate equation) value of q. Figure 1d shows the calculated relative errors as a function of time for the first system. As is clear in this figure, at the initial times of adsorption the value of the relative error is large and decreases with increasing time. This plot indicates that we cannot apply the MPFO model for the initial times of adsorption and the relative error of this model decreases as the system comes closer to equilibrium. This numerical analysis is in perfect agreement with our theoretical derivation of the MPFO equation from the SRT equation. Figure 1e shows the plot of ln(qe - q) + q/qe vs t for kinetic data close to equilibrium. The value of qe was adjusted until the intercept of the linear plot became equal to R′ (eq 30). The obtained values of qe and KM from Figure 1e are listed in Table 3. By comparison of the qe values in Tables 2 and 3, it can be observed that, by application of the MPFO equation for the kinetic data close to equilibrium, one can find more reasonable values of qe. By using the slope of Figure 1e, the Kls value was calculated by
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Figure 3. Application of numerically generated points (t, q) based on the SRT equation, ka ) 1.00 × 10-5 and kd ) 1.00 × 10-4 in different cases: (a) linear plot of the new initial rate equation for initial times of adsorption, (b) linear plot of q vs t for initial times of adsorption, (c) relative error of calculated q values based on the MPFO equation when the kinetic data in the whole range of time were applied, (d) linear plot of the MPFO equation when the system is close to equilibrium.
eq 26 and is reported in Table 2. The small difference between the calculated value of Kls by the initial rate equation (eq 12) and the MPFO equation (eq 30) and the original SRT value is due to the application of some approximations in derivation of eqs 12 and 30. Two other systems (data points of q and t) were generated by the SRT equation. For generation of data points in these systems the ka and kd values were chosen so that in one of them ka > kd and in the other one ka < kd. Figures 2a and 3a show plots of ln(KL2c02 - θ2) vs time (eq 12) for these two systems. Figures 2b and 3b show plots of q vs t (eq 9) for these two systems. Comparison of part a with part b of Figure 2 and also part a with part b of Figure 3 indicates that we can apply eq 12 in a longer range of time in comparison to eq 9. From the slopes of the linear plots in Figures 2a and 3a, the values of Kls were calculated and are presented in Table 2. The results are very close to the values of Kls obtained by the SRT equation (eq 33). Thus, again the accuracy of eq 12 was proved with these two systems. Now we apply the MPFO model to these new generated systems. On the basis of the plot of ln(qe - q) + q/qe vs time, in the whole range of time (graphs not presented here), the values of qe and KM were obtained and are listed in Table 3. Now using these constants and utilizing the MPFO rate equation, the values of q as a function of time were calculated, and then the relative errors of these results were obtained. Figures 2c and 3c show the relative errors of the MPFO equation for q values as a function of time for these two systems. Again these figures show that the
MPFO rate equation cannot be used for the initial times of adsorption. Figures 2d and 3d show plots of ln(qe - q) + q/qe vs t for the close to equilibrium kinetic data of new systems. The obtained values of qe and KM are presented in Table 3. The Kls values calculated from these results are listed in Table 2 too. These values of Kls are close to the obtained values from the SRT rate equation. As mentioned in the Theory section, if the value of KLc0 is much larger than θ (KLc0 . θ), then eq 12 will convert to eq 9. In this section, we investigate this using the generated data points (t, q) by SRT rate equation. The initial concentration of the solute is one of the effective factors in the above condition (KLc0 . θ). The first generated data points (KL ) 1) were studied in different initial concentrations of the solute (data not shown here). The calculation shows that, with increasing values of c0, the results of eq 12 come closer to the results of eq 9. This means that, when the value of c0 is large, eq 9 will be equivalent to eq 12 because the value of KLc0 is much larger than θ (KLc0 . θ). Next, we studied the effect of changing the values of θ on the range of time in which eqs 9 and 12 are applicable. Different values of θ were generated by changing the values of qm for the second system (KL > 1). The results show that, with increasing values of qm (decreasing θ), the results of eq 12 come closer to the results of eq 9. Then the effect of the values of ka and kd on the applicability range time of eqs 9 and 12 was studied. The results show that their
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Langmuir, Vol. 24, No. 20, 2008 11675
Figure 4. (a) Adsorption kinetics of methyl violet onto GAC: experimental data (open circles)16 and theoretical data based on the SRT equation (solid line). (b) Linear plot of the new initial rate equation for initial times of adsorption. (c) Linear plot of q vs t for initial times of adsorption. (d) Relative error of calculated q values based on the MPFO equation when the kinetic data in the whole range of time were applied.
absolute values are not important but their ratio (KL2 ) ka/kd) can change the applicable range time of eqs 9 and 12. Finally, we analyzed the experimental kinetic data of methyl violet adsorption onto granular activated carbon (GAC).16 The experimental data were fitted to the SRT equation (eq 33) by stochastic numerical simulation. The input data for this simulation are c0 ) 7.49 × 10-6 mol/L, qm ) 95.00 mg/g, Mw ) 393.95 g/mol (Mw is the molar mass of methyl violet), and cadsorbent ) 1.60 g/L. The values of ka and kd in eq 33 were adjusted till the best fit to the experimental kinetic data was obtained. The obtained values of ka, kd, and qe from the SRT equation are 1.53 × 10-4 min-1, 0.78 × 10-14 mol2/(L2 · min), and 1.81 mg/g, respectively. Figure 4a shows the experimental (open circles) and theoretical (SRT) (solid line) values of q as a function of time. Figure 4b shows good agreement between the experimental data and the results of eq 12 for the initial times of adsorption. The linear plot of q vs t on the basis of eq 9 is shown in Figure 4c. Comparison of parts b and c of Figure 4 shows that, for this system, eq 9 can be applied in the same range of time as eq 12 (because KLc0 . θ). Next the MPFO equation was applied for kinetic modeling of experimental data for the whole range of time. On the basis of the results of this modeling, the q values were calculated. Then the relative errors of these obtained q values were calculated and are plotted in Figure 4d as a function of time. As shown in Figure 4d the values of the relative errors are very high for the initial times of adsorption and decrease as time increases. This means that the MPFO rate equation can be used when the system is close to equilibrium. In summary, both experimental and numerically generated kinetic data show that the new kinetic equation, eq 12, is more appropriate
than the previous q vs t linear equation for kinetic modeling at initial times of adsorption. Also both experimental and numerically generated kinetic data are in perfect agreement with our theoretical derivation, which shows that the MPFO kinetic equation can be applied when the system is close to equilibrium.
Conclusion The adsorption kinetics at the solid/solution interface based on SRT has been studied at two limiting conditions, one at initial times of adsorption and the other one close to equilibrium. By considering some assumptions and approximations about the SRT equation, a new rate equation for initial times of adsorption was derived. It has been shown that usually we can apply the new equation for initial times of adsorption in a larger time range in comparison to the previous q vs t linear equation. For the first time, a theoretical background based on SRT was derived for the MPFO rate equation. It has shown theoretically that the MPFO equation can be derived from SRT when the system is close to equilibrium. On the basis of numerically generated points (q, t) using the SRT rate equation and also experimental data, it was shown that we cannot apply the MPFO equation for the whole time range of adsorption. Therefore, modeling of kinetic data by the MPFO equation in the whole range of time creates a large error at initial times of adsorption.
Glossary A
Nomenclature a function which is defined by eq 16 (mol/L)
11676 Langmuir, Vol. 24, No. 20, 2008
Ae c c0 cadsorbent ce k ka kd KL Kls Kls′ KM Mw q qcalcd qe qexptl
equilibrium value of A (mol/L) bulk concentration of the solute (mol/L) initial concentration of the solute in the bulk (mol/L) concentration of the adsorbent (g/L) bulk equilibrium concentration of the solute (mol/L) Boltzmann constant (J/K) rate constant of adsorption (min-1) rate constant of desorption [mol2/(L2 · min)] Langmuir constant (L/mol) constant in eq 7 [L/(mol · min)] equilibrium rate of adsorption at the solid/solution interface (min-1) MPFO constant (min-1) molar mass (g/mol) amount of the adspecies (mg/g) amount of the adspecies calculated by the MPFO kinetic model (mg/g) amount of the adspecies at equilibrium (mg/g) experimental or generated value of q (mg/g)
Azizian and Bashiri
qm qs RE t t1 T R R′ β µb µb° µs θ θ1 θe LA802288P
maximum value of the amount of the adspecies (mg/g) molecular partition function relative error time (min) time after which the MPFO rate equation can be applied (min) absolute temperature (K) MPFO constant which is defined by eq 29 constant in eq 30 constant in eq 13 (mol/L) chemical potential of the adsorbate in the bulk phase standard chemical potential of the adsorbate in the bulk phase chemical potential of the adsorbate on the surface phase fractional surface coverage fractional surface coverage at time t1 fractional surface coverage at equilibrium
