Kinetic modeling of liquid-phase adsorption of reactive dyes on activated carbon

Kinetic modeling of liquid-phase adsorption of reactive dyes on activated carbon

Journal of Colloid and Interface Science 287 (2005) 25–34 www.elsevier.com/locate/jcis Kinetic modeling of liquid-phase adsorption of reactive dyes o...

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Journal of Colloid and Interface Science 287 (2005) 25–34 www.elsevier.com/locate/jcis

Kinetic modeling of liquid-phase adsorption of reactive dyes on activated carbon Xiaoyan Yang ∗ , Bushra Al-Duri School of Chemical Engineering, the University of Birmingham, Birmingham, B15 2TT, UK Received 25 November 2004; accepted 25 January 2005 Available online 10 March 2005

Abstract In this paper, adsorption equilibrium and kinetics of three reactive dyes from their single-component aqueous solutions onto activated carbon were studied in a batch reactor. Effects of the initial concentration and adsorbent particle size on adsorption rate were investigated Adsorption equilibrium data were then correlated with several well-known equilibrium isotherm models. The kinetic data were fitted using the pseudo-first-order equation, the pseudo-second-order equation, and the intraparticle diffusion model. The respective characteristic rate constants were presented. A new adsorption rate model based on the pseudo-first-order equation has been proposed to describe the experimental data over the whole adsorption process. The results show that the modified pseudo-first-order kinetic model generates the best agreement with the experimental data for the three single-component adsorption systems.  2005 Elsevier Inc. All rights reserved. Keywords: Adsorption; Equilibrium; Kinetics; Pseudo-first-order equation; Pseudo-second-order equation; Intraparticle diffusion model

1. Introduction Adsorption by activated carbon has been used extensively in dye house wastewater treatment, either independently or coupled with biological degradation [1]. Though many lowcost materials have been tested for the removal of different dyes from their aqueous solutions, such as clay [2], sawdust [3], chitosan [4], and peat [5], adsorption by activated carbon remains one of the most efficient techniques in dye house wastewater treatment, especially as a final polishing step before discharging or recycling the treated wastewater. This is mainly due to the well-developed porous internal structure of activated carbon and its various surface functional groups, and hence its remarkable adsorption properties. In this regard, activated carbon has been evaluated extensively for the treatment of aqueous effluents containing different classes of dyes [3,6–8]. * Corresponding author. Present address: School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Fax: +44-131-451-3129. E-mail address: [email protected] (X. Yang).

0021-9797/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.01.093

The study of adsorption equilibrium and kinetics is essential in supplying the fundamental information required for the design and operation of adsorption equipments for wastewater treatment. Different models have been put forward to describe or predict the adsorption kinetics. Tworesistance models, such as the film–pore model [9], the film–solid model [10], and the branched pore diffusion model [11], provide a detailed description of the adsorption process. Their solutions are usually in the form q(r, t), which indicates that the solid phase concentration q is a function of both the spatial location inside the particle r and the time t. However, these models contain partial differential equations and it usually requires extensive numerical development and high computational time to obtain the most appropriate model parameters and the corresponding solution q(r, t) at these parameter values. This might be regarded as impractical for plant simulations, where simple and explicit relationships between the adsorption performance and operating conditions are preferable. Compared with the two-resistance models, lumped kinetic models showing how the spatially averaged solid phase concentration qt changes with the adsorption time are much

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simpler and easier to apply. Two well-known models in this category are the pseudo-first-order and pseudo-secondorder rate equations. Both models assume that the difference between the average solid phase concentration qt and the equilibrium concentration qe is the driving force for adsorption and the overall adsorption rate is proportional to either the driving force (as in the pseudo-first-order equation) or the square of the driving force (as in the pseudo-secondorder equation). The applicability of the two models can be confirmed by a linear plot of ln(qe − qt ) against t and a linear plot of t/qt against t, respectively. Both equations have been widely employed to test the adsorption mechanisms [4, 12–14]. Another commonly used lumped kinetic model is the intraparticle diffusion model, which is derived from Fick’s law and describes the adsorption process in terms of the qt –t 0.5 relationship. If the adsorption process follows the intraparticle diffusion model, a plot of qt against t 0.5 should be a straight line. The intraparticle diffusion model has also been applied to various adsorption systems [4,14,15] and it has been found that the model could account for the adsorption mechanism in many well-stirred adsorption systems. Undoubtedly, the lumped kinetic models have provided a simple and satisfactory explanation to the whole adsorption process for many adsorption systems. Nevertheless, for many other adsorption systems, the plot of either ln(qe − qt ) against t, t/qt against t, or qt against t 0.5 may be multilinear [4,15,16]. To tackle this problem, it is common practice to segment the plot into two or more straight lines and to suggest that different adsorption mechanism controls each step represented by each straight line. However, segmenting the plot into two or more straight lines of different slopes and different intercepts can be quite “artificial” as it is often up to the researcher to decide where to segment the multilinear plot. Though it may help understand the adsorption mechanism to a certain extent, the practical significance of such an approach is quite limited. One indication of the multilinearity of the plot is that the rate constants in the kinetic models may not be invariable. They may vary with time, or more specifically with the solid phase concentration qt . This leads to the idea of incorporating a qt -dependent rate constant into the lumped kinetic models. The objective of this paper is twofold: first, to test the validity of the pseudo-first-order equation, the pseudo-secondorder equation and the intraparticle diffusion model for the adsorption of three reactive dyes in their single component systems onto activated carbon; second, to propose a new lumped kinetic model by incorporating a qt -dependent rate constant into the pseudo-first-order equation. The new model is defined as the modified pseudo-first-order kinetic model. The outline of the paper is as follows: the theoretical background, i.e., adsorption equilibrium isotherm models and four different kinetic models, are presented. Then, the models are fitted to the experimental data and their characteristic constants (equilibrium constants and rate constants) derived.

This is followed by the results and discussion. Several conclusions are drawn in the end.

2. Theoretical background 2.1. Adsorption equilibrium Equilibrium study provides fundamental information required to evaluate the affinity or capacity of an adsorbent, which is one of the most important criteria in selecting a suitable adsorbent [17]. Meanwhile, equilibrium behavior of an adsorption system is an essential prerequisite during mathematical modeling of the adsorption kinetics. A clear review of the various equilibrium isotherms and their applications is in literature [18]. Among the available adsorption equilibrium isotherm models, the most generalized model is a correlative equation proposed by Fritz and Schlünder [19] for the prediction of the adsorption of multicomponent systems, which has the form b

qi =

ki0 Ci i0 ,  b ci + nj=1 aij Cj ij

(1)

where qi and Ci are the equilibrium concentration of component i in solid phase and liquid phase, respectively. For an adsorption system containing only component i, Cj = 0 (j = i), Eq. (1) is reduced to the Fritz–Schlünder isotherm for single-component systems, q=

k0 C b0 , c + aC b

(2)

which is usually written as q=

ks C b1 , 1 + as C b2

(3)

where ks (mg/g)(mg/L)−b1 , as (mg/L)−b2 , b1 (—), and b2 (—) are the Fritz–Schlünder isotherm constants. A scrutiny into Eq. (2) reveals that this equation can be further reduced to several well-known equilibrium isotherm equations under certain conditions. For example, assuming b0 = b = c = 1 and replacing k0 and a by the Langmuir isotherm constants KL (L/g) and aL (L/mg), respectively, Eq. (2) becomes the Langmuir isotherm q=

KL C . 1 + aL C

(4)

The Freundlich isotherm can be obtained when assuming c = 0 in Eq. (2), k0 b0 −b C , a or more specifically

q=

(5)

q = KF C nF ,

(6)

X. Yang, B. Al-Duri / Journal of Colloid and Interface Science 287 (2005) 25–34

where KF (mg/g)(L/mg)nF and nF (—) are the Freundlich isotherm constants. The Redlich–Peterson isotherm comes to light when b0 = c = 1 in Eq. (2), k0 C , 1 + aC b which is usually known as

q=

q=

KR C , 1 + bR C β

(7)

(8)

where KR (L/g), bR (mg/L)−β , and β (—) are the Redlich– Peterson isotherm constants. Being an empirical correlation of the q–C relationship, the Fritz–Schlünder isotherm (Eqs. (2) and (3)) does not have a theoretical foundation. However, the Fritz–Schlünder isotherm has higher flexibility to fit the experimental data since it contains more parameters than any other equilibrium isotherm does. Thus, it is not surprising that it has been found to be able to fit experimental data over a large concentration range [11,20]. 2.2. Adsorption kinetics As aforementioned, a lumped analysis of adsorption rate is sufficient to practical operation from a system design point of view. The commonly employed lumped kinetic models, namely, (a) the intraparticle diffusion model, (b) the pseudofirst-order equation, and (c) the pseudo-second-order equation are presented below. 2.2.1. Intraparticle diffusion model Intraparticle diffusion model assumes that the film diffusion is negligible and intraparticle diffusion is the only ratecontrolling step, which is usually true for well-mixed solutions. The intraparticle diffusion model is a single-resistance model in nature and can be derived from Fick’s second law under two assumptions [21]: first, the intraparticle diffusivity D is constant; second, the uptake of sorbate by the adsorbent is small relative to the total quantity of sorbate present in the solution. The mathematical expression thus obtained for the intraparticle diffusion model is qt ≈ kp t 0.5 ,

(9)

where kp (mg/g min−0.5 ) is defined as the intraparticle diffusion rate constant and is related to the intraparticle diffusivity in the following way,  6qe D , kp = (10) R π where R (cm) is the particle radius and qe (mg/g) is the solid phase concentration at equilibrium. Equation (9) indicates that a plot of the average particle loading, qt (mg/g), versus the square root of time, t 0.5 , would yield a straight line passing through the origin if the adsorption process obeyed the intraparticle diffusion model.

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The slope of the straight line equals to kp , the intraparticle diffusion rate constant. 2.2.2. Pseudo-first-order and pseudo-second-order kinetic models The pseudo-first-order and pseudo-second-order kinetic models assume that adsorption is a pseudo-chemical reaction process and the adsorption rate can be determined respectively by the first-order and second-order reaction rate equations, dqt = k1 (qe − qt ), dt dqt = k2 (qe − qt )2 , dt

(11) (12)

where qe (mg g−1 ) is the solid phase concentration at equilibrium, qt (mg g−1 ) is the average solid phase concentration at time t (min), and k1 (min−1 ) and k2 (g mg−1 min−1 ) are the pseudo-first-order and pseudo-second-order rate constants, respectively. The above equations represent initial value problems and have analytical solutions when combined with the initial condition t = 0, qt = 0. The solutions for Eqs. (11) and (12) are as follows: ln(qe − qt ) = ln(qe ) − k1 t, t 1 t = + . 2 qt qe k2 q e

(13) (14)

If the adsorption follows the pseudo-first-order rate equation, a plot of ln(qe − qt ) against time t should be a straight line. Similarly, t/qt should change linearly with time t if the adsorption process obeys the pseudo-second-order rate equation. Available studies have shown that the pseudo-secondorder rate equation is a reasonably good fit of data over the entire fractional approach to equilibrium and therefore has been employed extensively in the study of adsorption kinetics [4,12–14]. However, it is not uncommon to observe multilinearity on the ln(qe − qt )–t plot or t/qt –t plot. The trend is usually such that the rate constant decreases with time, or more specifically decreases with increasing solid phase concentration. 2.2.3. Modified pseudo-first-order kinetic model In this paper, a modified pseudo-first-order kinetic model is proposed which may prove to be especially useful when the other three kinetic models (the pseudo-first-order equation, the pseudo-second-order equation, and the intraparticle diffusion model) fail to generate good agreement with the experiment. To set up the new model, the pseudo-first-order equation represented by Eq. (11) is modified through the modification of its rate constant. Denote the rate constant in the modified pseudo-first-order rate equation by K1 , the following equation is proposed: qe k1 = K1 . (15) qt

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As qt < qe , the above equation implies that the rate constant k1 is minimum when equilibrium is reached. The modified pseudo-first-order rate equation can be derived as follows: qe dqt = K1 (qe − qt ). (16) dt qt

(a)

Equation (16) can be rearranged into −dqt +

qe dqt = K1 qe dt. (qe − qt )

(17)

Integrate Eq. (17) over time t ∈ (0, t), during which the solid phase concentration increases from zero to qt , the following algebraic equation can be obtained: qt + ln(qe − qt ) = ln(qe ) − K1 t. (18) qe

(b)

If the adsorption process follows the modified pseudo-firstorder kinetic model represented by Eq. (18), a plot of qt /qe + ln(qe − qt ) against t should be a straight line. (c)

3. Experimental

Fig. 1. Molecular structures of (a) RN, (b) RR, and (c) RY.

3.1. Materials/analytical methods Activated carbon Filtrasorb 400 (F400) produced by Calgon Carbon Corporation was selected as the adsorbent. Three reactive dyes provided by Ciba-Geigy, namely Cibacron Red F-B (abbreviated as RR), Cibacron Navy F-G (abbreviated as RN), and Cibacron Yellow F-3R (abbreviated as RY) were chosen as the adsorbates, due to their extensive use in the textile industry. Their molecular structures (provided by the producer) are shown in Figs. 1a–1c. Prior to its use, the activated carbon was crushed and screened to a series of particle sizes (0.0412, 0.0536, 0.0816, and 0.105 cm). It was then washed thoroughly in distilled water to remove fines. Afterwards it was dried at 110 ◦ C for 24 h and stored in a desiccator. Dye concentrations in aqueous solutions were measured using a spectrophotometer. According to the standard method suggested by the National Rivers Authority [1], all samples were filtered through a 0.45-µm membrane filter paper before measurement in order to remove the carbon fines. 3.2. Adsorption isotherm study Equilibrium isotherms were determined at 25 ◦ C, by shaking fixed masses of activated carbon with 200 ml dye solutions of consecutively increasing concentrations in capped conical flasks. Shaking was carried out for 25 days to allow equilibrium to be attained. 3.3. Adsorption rate study The operating conditions which influence adsorption rate include initial concentration C0 (mg/L), mass of activated

carbon m (g), agitation speed A (rpm), and particle size of activated carbon dp (cm). In this study, the effect of the particle size and the initial concentration on adsorption kinetics was investigated, as they are more directly connected with intraparticle diffusion than the mass of activated carbon and agitating speed are, the latter playing a more important role in the hydrodynamic condition of the liquid phase. Kinetic studies were carried out in a 3-L-glass vessel. The vessel was equipped with six stainless steel baffles distributed evenly around the circumference, and a sampling port with a rubber septum which allowed for the withdrawal of samples from a fixed point near the centre of the vessel without interrupting the experiments. The solution was stirred by a six-flat-blade impeller driven by an electric motor. All batch experiments lasted about 30 h and samples were taken at predetermined time intervals. In all the experiments, the volume of solution was 2.50 L, the mass of activated carbon was 7.50 g, and the agitation speed was 400 rpm. Carbon particles of four different diameters were used for the study of particle size effect; no less than four initial concentrations for each dye/F400 system were employed for the study of initial concentration effect on adsorption.

4. Results and discussion 4.1. Adsorption equilibrium The adsorption equilibrium data of the three adsorption systems were fitted with the Langmuir equation, the Freundlich equation, the Redlich–Peterson equation, and the Fritz–Schlünder equation. In order to get the best fitting,

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Table 1 Parameter values from fitting the adsorption equilibrium data with various isotherm models for three reactive dyes Isotherm model

RR

RN

RY

Parameter values

Nonnormalized RMS

Parameter values

Nonnormalized RMS

Parameter values

Nonnormalized RMS

Langmuir

KL = 1.79 aL = 0.0189 KL /aL = 94.7

4.10

KL = 1.92 aL = 0.0227 KL /aL = 84.6

5.53

KL = 26.7 aL = 0.134 KL /aL = 199

15.6

Freundlich

KF = 20.0 nF = 0.217

8.70

KF = 19.6 nF = 0.208

7.16

KF = 67.7 nF = 0.161

30.6

Redlich–Peterson

KR = 3.66 bR = 0.0761 β = 0.906 KR /bR = 48.1

1.87

KR = 7.83 bR = 0.226 β = 0.874 KR /bR = 34.6

2.81

KR = 28.6 bR = 0.160 β = 0.983 KR /bR = 179

15.4

Fritz–Schlünder

ks = 5.51 as = 0.0759 b1 = 0.770 b2 = 0.725 ks /as = 72.6

1.56

ks = 9.23 as = 0.180 b1 = 0.716 b2 = 0.639 ks /as = 51.3

2.59

ks = 1.61 as = 0.0134 b1 = 3.31 b2 = 3.23 ks /as = 120

8.73

Newton optimization procedure was employed to minimize the root mean square (RMS) of the residuals between theoretical and experimental solid phase concentrations, denoted by qe,cal and qe,exp , respectively. There are two kinds of RMS; the normalized RMS weights all points equally, while the nonnormalized RMS weights the actual error at all points,  N  Normalized RMS =  (1 − qei,cal /qei,exp )2 /N , (19) i=1

 N  Nonnormalized RMS =  (qei,cal − qei,exp )2 /N,

(20)

i=1

where N is the number of points on a fitted experimental curve. It has been noticed that in this study, the result of using normalized RMS is to achieve better fitting for the concentration range of Ce < 16 mg/L at the expense of the good fitting for a much wider concentration range 16 mg/L < Ce < 2000 mg/L. Thus, the nonnormalized RMS is more suitable as a criterion than the normalized RMS. The results of the fitting procedure are presented in Table 1. Table 1 shows that, for RR/F400 and RN/F400 systems, both the Redlich–Peterson isotherm and the Fritz–Schlünder isotherm could yield good fits over the whole concentration range, while the Langmuir isotherm and the Freundlich isotherm gave a poor fit. For RY/F400 system, the Fritz– Schlünder isotherm is the only model to produce acceptable fit to the experimental data. Fig. 2 compares the experimentally determined adsorption equilibrium data with the model prediction by the Fritz– Schlünder isotherm. The excellent fitting of Fritz–Schlünder isotherm can be due to the fact that it contains more parameters than the other equilibrium isotherms do. This implies that it has more degree of freedom to fit the experimental

Fig. 2. Equilibrium isotherms for three single-component adsorption systems.

data by changing another parameter if one parameter has ceased to generate good agreement between the model and the experiment. 4.2. Adsorption kinetics The adsorption rate denoted by dq/dt shows how much adsorbate can be adsorbed from the liquid phase onto the adsorbent within a unit time. In a diagram depicting the qt –t relationship, the slope at each point of the curve represents the instantaneous adsorption rate dq/dt. However, the qt –t plot does not shed much light on how close the adsorption process at time t is towards its equilibrium state, which is also an important concern for adsorption operation. For a batch adsorption operation, the temporal approach to equilibrium can be illustrated by a plot of the fractional uptake f against time t, where f = qt /qe . The effect of the initial bulk concentration on the adsorption kinetics for RN/F400 system is shown in Fig. 3 in terms of both the qt –t relationship and the f –t relationship. Fig. 4

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(a)

(a)

(b)

(b)

Fig. 3. Effect of initial concentration on the adsorption of RN onto F400 (dp = 0.0536 cm) in terms of (a) qt –t relationship; (b) f –t relationship.

Fig. 4. Effect of adsorbent size on the adsorption of RY onto F400 (C0 = 88.4 mg/L) in terms of (a) qt –t relationship; (b) f –t relationship.

displays the effect of the adsorbent size on the adsorption kinetics of RY/F400 system. The effect of the initial concentration and adsorbent size on the adsorption of the other reactive dyes onto F400 has a similar trend as shown in Figs. 3 and 4. Figs. 3 and 4 reveal that the adsorption rate (dqt /dt) decreases with time until it gradually approaches the equilibrium state due to the continuous decease in the driving force (qe − qt ). They also demonstrate that the adsorbate uptake qt increases with increasing initial concentration while decreases with increasing adsorbent size (Figs. 3a and 4a). On the other hand, Fig. 3b shows that the fractional uptake f decreases with increasing initial concentration, although the tendency is not so obvious within the high concentration range. The fractional uptake has also been shown to decrease with increasing adsorbent size (Fig. 4b).

respectively. The derived rate constants together with the correlation coefficient R 2 for the three adsorption systems, i.e., RN/F400, RR/F400 and RY/F00, have been listed in Tables 2–4. Figs. 5–7 display the best-fitting results by the modified pseudo-first-order rate equation, also shown in the figures are the experimental data. Several conclusions can be drawn from Tables 2–4 and Figs. 5–7. (1) Among the four kinetic models, the modified pseudofirst-order equation generates the best fit to the experimental data of the three investigated adsorption systems. All the correlation coefficients obtained are bigger than 0.99, with only one exception R 2 = 0.986 for the RY/F00 system at dp = 0.0536 cm and C0 = 60.3 mg/L. This indicates that the modified pseudo-first-order equation is potentially a generalized kinetic model for adsorption study. However, there does not appear to exist a general “second best” model to describe all the adsorption systems. For RN/F400, the second best model to generate a good fit to the experiment is the intraparticle diffusion model, followed by the pseudo-second-order kinetic model and lastly the pseudo-first-order kinetic model. For RR/F400, the intraparticle diffusion model generates a better agreement with the experiment than the pseudo-second-order model

4.2.1. Comparison of the applicability of different kinetic models In order to test the applicability of the four different kinetic models, namely the pseudo-first-order, the pseudosecond-order, the intraparticle diffusion model, and the modified pseudo-first-order kinetic model, the experimental data were correlated with the linear forms of the four models,

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Table 2 Results from linear regression of the adsorption rate experiments for RN/F400 system dp (cm)

C0 (mg/L)

First order

0.0536

0.0412 0.0816 0.105

Modified first order

k1 (1/min)

R2

K1 (1/min)

R2

13.2 43.3 76.1 105.8 163.5

8.42E−4 4.42E−4 3.07E−4 2.96E−4 2.66E−4

0.967 0.945 0.943 0.939 0.904

4.28E−4 1.45E−4 7.63E−5 7.01E−5 6.12E−5

105.8 105.8 105.8

3.57E−4 2.42E−4 1.97E−4

0.937 0.947 0.955

1.02E−4 4.83E−5 3.27E−5

Second order

Intraparticle diffusion

k2 (g/mg min)

R2

kp (mg/g min0.5 )

R2

0.998 0.996 0.998 0.999 0.993

1.00E−3 3.66E−4 2.54E−4 1.82E−4 2.02E−4

0.977 0.969 0.965 0.971 0.978

8.75E−2 1.89E−1 2.44E−1 3.15E−1 4.35E−1

0.980 0.993 0.996 0.996 0.980

0.999 0.999 0.996

2.08E−4 1.99E−4 2.11E−4

0.969 0.957 0.940

3.75E−1 2.64E−1 2.18E−1

0.991 0.998 0.995

Table 3 Results from linear regression of the adsorption rate experiments for RR/F400 system dp (cm)

C0 (mg/L)

First order

0.0412

0.0536 0.0816 0.105

Modified first order

k1 (1/min)

R2

K1 (1/min)

R2

12.0 40.1 69.3 97.9 151.6

1.74E−3 1.08E−3 5.62E−4 4.45E−4 3.95E−4

0.969 0.981 0.945 0.944 0.920

1.19E−3 6.41E−4 2.29E−4 1.53E−4 1.21E−4

97.9 97.9 97.9

3.07E−4 2.55E−4 2.15E−4

0.924 0.916 0.931

8.00E−5 5.59E−5 3.93E−5

Second order

Intraparticle diffusion

k2 (g/mg min)

R2

kp (mg/g min0.5 )

R2

0.993 0.992 0.999 0.998 0.999

1.66E−3 3.23E−4 2.42E−4 2.33E−4 1.95E−4

0.991 0.978 0.974 0.955 0.965

0.110 0.279 0.344 0.406 0.548

0.900 0.971 0.979 0.977 0.973

0.999 0.998 0.998

3.02E−4 3.13E−4 2.92E−4

0.957 0.959 0.944

0.316 0.269 0.223

0.978 0.986 0.996

Table 4 Results from linear regression of the adsorption rate experiments for RY/F400 system dp (cm)

C0 (mg/L)

First order

0.0536

0.0412 0.0536 0.0816 0.105

Modified first order

k1 (1/min)

R2

K1 (1/min)

R2

35.4 60.3 87.2 131.2

2.48E−3 1.99E−3 1.25E−3 1.07E−3

0.968 0.994 0.977 0.974

1.95E−3 1.43E−3 7.55E−4 5.96E−4

88.4 88.4 88.4 88.4

1.77E−3 1.18E−3 8.06E−4 5.44E−4

0.991 0.986 0.976 0.966

1.27E−3 7.15E−4 3.98E−4 2.11E−4

except at low concentrations (C0 = 12.0 and 40.1 mg/L). For RY/F400, the pseudo-first-order kinetic model and the pseudo-second-order kinetic model produce similar correlation coefficients, while the correlation coefficients for the intraparticle diffusion model range from 0.838 to 0.995, indicating the intraparticle diffusion model has a limited applicability to the RY/F400 adsorption system. It is worth noting that, the low correlation coefficients shown in Tables 3 and 4 do not necessarily mean that the intraparticle diffusion process is not the rate-controlling step. It is a mere indication that the intraparticle diffusion model does not apply to the investigated adsorption systems. This may be due to the following two reasons: first, the intraparticle diffusion model assumes infinite solution volume

Second order

Intraparticle diffusion

k2 (g/mg min)

R2

kp (mg/g min0.5 )

R2

0.995 0.986 0.992 0.993

5.42E−4 1.93E−4 1.36E−4 9.45E−5

0.999 0.992 0.988 0.981

0.345 0.530 0.681 0.966

0.838 0.950 0.967 0.977

0.994 0.993 0.995 0.994

1.71E−4 1.38E−4 1.36E−4 1.56E−4

0.990 0.980 0.970 0.961

0.767 0.665 0.561 0.451

0.933 0.976 0.990 0.995

control, which implies that there is no change in solution concentration during the approach to equilibrium. This is only achieved when the product of the solution volume and solution concentration greatly exceeds the product of the adsorbent mass and the equilibrium sorption capacity of the adsorbent. However, this is not the case for the investigated adsorption systems. Second, the intraparticle diffusion model is derived assuming a constant diffusivity. However, available studies [10,11] have shown that the diffusivity of the investigated adsorption systems depends on the solid phase concentration to a large extent. The two factors may have contributed to the less than satisfactory agreement between the experiment and the prediction by the intraparticle diffusion model.

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(a)

(a)

(b)

(b)

Fig. 5. Test of the modified pseudo-first-order kinetic model for adsorption of RN (a) at different initial concentrations; (b) on different sizes of F400.

Fig. 6. Test of the modified pseudo-first-order kinetic model for adsorption of RR (a) at different initial concentrations; (b) on different sizes of F400.

(2) Regardless of the correlation coefficients, the rate constants k1 and K1 decrease with increasing initial concentration for all the three adsorption systems. However, the rate constant of the intraparticle diffusion model kp paints a different picture, with kp increasing with increasing initial concentration. The reverse effect of the initial concentration on k1 and kp has also been reported for other adsorption systems [4,22], though no explanation has been provided. The following analysis reveals that the seemingly contradictory trends are in fact compatible with each other. The rate constant of intraparticle diffusion model kp has a unit of mg/(g min0.5 ) and is a good representation of the adsorption rate dq/dt (mg/g min). As can be seen from Eq. (10), kp is proportional to both qe and D 1/2 . When the initial dye

concentration C0 is increased, the equilibrium concentration qe will be increased accordingly. In addition, the intraparticle diffusivity D has been found to increase with the initial concentration C0 [6,10,11,23]. Therefore, the increasing kp with increasing initial concentration C0 can be attributed to the increase in both qe and D 1/2 . Apart from dqt /dt, there exists another measure of the rate of adsorption—the half adsorption time t1/2 , which is the time required for the adsorbent to take up half as much adsorbate as it does at equilibrium [12]. In general, the time required for the adsorbent to take up a fraction f ∈ (0, 1) as much adsorbate as it does at equilibrium, tf , can be calculated from Eq. (13) and Eq. (18) for the pseudo-first-order model and the modified pseudo-first-order model. The rate

X. Yang, B. Al-Duri / Journal of Colloid and Interface Science 287 (2005) 25–34

(a)

(b) Fig. 7. Test of the modified pseudo-first-order kinetic model for adsorption of RY (a) at different initial concentrations; (b) on different sizes of F400.

constants k1 and K1 can then be written as a function of tf : ln(1 − f ) (21) , tf f + ln(1 − f ) . K1 = − (22) tf It is clear that both k1 and K1 are reciprocal of tf and have a unit of min−1 . A larger k1 or K1 implies that it will take shorter time for the adsorption system to reach the same fractional uptake. Thereby, the trend that k1 and K1 decrease with increasing initial concentration only reveals the fact that it is faster for an adsorption system with a lower initial concentration to reach a specific fractional uptake (as shown in Fig. 3b), though its adsorption rate within the same period of time will be slower, as indicated by kp . k1 = −

33

Tables 2–4 also show that the rate constant k2 decreases with increasing initial concentration, although the trend is not so obvious at very high initial concentration for RN/F400 adsorption system. This conclusion is in consistent with those observed for other adsorption systems [4,12,22]. (3) For the three adsorption systems, the rate constants k1 , K1 , and kp all increase with decreasing particle size. As the influence of particle size on the equilibrium concentration is insignificant [23], qe can be considered constant for adsorbent of different particle sizes at the same initial concentration. Equation (10) clearly shows that kp depends reciprocally on the adsorbent radius. The increase of rate constants k1 and K1 with decreasing particle size is also expected, as smaller particles have larger external surface area which increases the opportunities for collision between the adsorbate and the adsorbent, causing the adsorption rate on smaller particles to be faster than on larger particles. This results in a smaller tf for smaller adsorbent particles, and consequently k1 and K1 increase. The rate constant k2 does not display a clear pattern with the increasing particle size. This may be due to the low correlation coefficients resulted from the linear regression based on the pseudo-second-order kinetic model. For example, the value of R 2 is as low as 0.94 at dp = 0.105 cm for RN/F400 and RR/F400 adsorption systems. This implies that the correlation equation may not be applicable. (4) For all the three adsorption systems, the rate constant K1 is smaller than the corresponding rate constant k1 . This is self-explanatory from Eq. (15), k1 = K1 qe /qt . 4.2.2. Comparison of K1 between different adsorption systems The physiochemical properties of the adsorbate and the adsorbent have an important role to play in determining adsorption equilibrium and adsorption kinetics. As shown in Fig. 1, RN, RR, and RY have different molecular size and different numbers of functional groups. Therefore, it is expected that the adsorption behavior of the three reactive dyes and the related kinetic parameters would differ from each other. As the modified pseudo-first-order kinetic model fits the experiment best, attention is focused on the analysis of its rate constant. Figs. 8a and 8b show the dependence of the rate constant K1 of the three adsorption systems on the initial molar concentration M0 and particle size, respectively. The two figures also exhibit the difference in K1 between the three reactive dyes. It is interesting to observe that, at the same initial molar concentration and the same particle size, the rate constant K1 of the RY/F400 system is much larger than that of RR/F400 system, which in turn is slightly larger than that of RN/F400 system. It has been noted that activated carbon of 0.0412 cm was used in the study of the effect of the initial concentration for RR/F400 system. When taking into consideration the effect of particle size on adsorption rate (see Fig. 8b and Tables 2–4), the rate constant of the RR/F400 system at dp = 0.0536 still lies between those of RN/F400 system and

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(a)

of the three adsorption systems can be best fitted by the four-parameter Fritz–Schlünder isotherm model. A modified pseudo-first-order kinetic model has been proposed and compared to three widely used kinetic models, namely the pseudo-first-order rate equation, the pseudo-second-order rate equation and the intraparticle diffusion model. The results show that the modified pseudo-first-order kinetic model generates the best fit to all the experimental data, indicating it is potentially a generalized kinetic model for adsorption study. The rate constant of the modified pseudo-first-order kinetic model has been found to decrease with both increasing initial concentration and increasing adsorbent size. It has also been found that the rate constant of RY/F400 system is much larger than that of RR/F400 system, which in turn is slightly larger than that of RN/F400 system. This has been attributed to the difference in molecular size and polarity of the three reactive dyes.

Acknowledgments The authors thank the board of CVCP and the School of Chemical Engineering, the University of Birmingham for their financial support.

References

(b) Fig. 8. Dependence of K1 on (a) the initial concentration and (b) the adsorbent size for the three adsorption systems.

RY/F400 system. Such trend can be accounted for by the difference in the physiochemical properties of RY, RR and RN. As can be seen from Fig. 1, RY has the smallest molecular size and the least number of polar functional groups among the three dyes. Therefore, RY possesses higher mobility and is more easily adsorbed by activated carbon, which is more effective in adsorbing non-polar organics. Compared with RY/F400 system, the K1 values of RR/F400 and RN/F400 are rather close. The similarity in K1 between RR/F400 and RN/F400 can again be attributed to the similarity in the molecular structure and polarity between the two reactive dyes.

5. Conclusions The adsorption equilibrium and kinetics of three adsorption systems (RN/F400, RR/F400 and RY/F400) have been studied. It is found that the adsorption equilibrium

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