Langmuir 2008, 24, 11625-11630
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From Langmuir Kinetics to First- and Second-Order Rate Equations for Adsorption Yu Liu* and Liang Shen DiVision of EnVironmental and Water Resources Engineering, School of CiVil and EnVironmental Engineering, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798, Singapore ReceiVed June 11, 2008. ReVised Manuscript ReceiVed July 24, 2008 So far, the first- and second-order kinetic equations have been most frequently employed to interpret adsorption data obtained under various conditions, whereas the theoretical origins of these two equations still remain unknown. Using the Langmuir kinetics as a theoretical basis, this study showed that the Langmuir kinetics can be transformed to a polynomial expression of dθt/dt ) k1(θe - θt) + k2(θe - θt)2, a varying-order rate equation. The sufficient and necessary conditions for simplification of the Langmuir kinetics to the first- and second-order rate equations were put forward, which suggested that the relative magnitude of θe over k1/k2 governs the simplification of the Langmuir kinetics. In cases where k1/k2 is greater than θe or k1/k2 is very close to θe, adsorption kinetics would be reasonably described by the first-order rate equation, whereas the Langmuir kinetics would be reduced to the second-order equation only at k1/k2 , θe. It was further demonstrated that both θe and k1/k2 are the function of initial adsorbate concentration (C0) at a given dosage of adsorbent, indicating that simplification of the Langmuir kinetics indeed is determined by C0. Detailed C0-depedent boundary conditions for simplifying the Langmuir kinetics were also established and were verified by experimental data.
Introduction The first- and second-order rate equations have been widely used to describe adsorption data obtained under nonequilibrium conditions. The first-order rate equation or the so-called Lagergren equation is commonly expressed as dqt/dt ) k′1(qe - qt), in which qt is the amount of adsorbate adsorbed at time t, qe is its value at equilibrium, and k′1 is a constant. The first-order rate equation indeed is in line with the concept of linear driving force. The second-order rate equation was first proposed by Blanchard et al.,1 and it has been frequently employed to analyze adsorption data obtained from various experiments with different types of adsorbates and adsorbents as reviewed by Ho et al.:2 dqt/dt ) k′2(qe - qt)2, in which k′2 is a constant. In nearly all kinetic studies of adsorption, both first- and secondorder rate equations have been commonly employed in parallel, and one is often claimed to be better than another according to a marginal difference in correlation coefficient.3-12 As noted by Rudzinski and Plazinski,13 in the past decades no attempts were made to clearly explain the theoretical origins of these two empirical rate equations for adsorption. In fact, current understanding of adsorption kinetics is much less than the theoretical description of adsorption equilibrium. The present study thus * Corresponding author. E-mail:
[email protected]. (1) Blanchard, G.; Maunaye, M.; Martin, G. Water Res. 1984, 18, 1501–1507. (2) Ho, Y. S.; Ng, J. C. Y.; McKay, G. Sep. Purif. Methods 2000, 29, 189–232. (3) Liu, Y.; Yang, S. F.; Xu, H.; Woon, K. H.; Lin, Y. M.; Tay, J. H. Process Biochem. 2003, 38, 997–1001. (4) Krishnan, K. A.; Anirudhan, A. S. T. S. J. Chem. Technol. Biotechnol. 2003, 78, 642–653. (5) Aksu, Z.; Kabasakal, E. Sep. Purif. Technol. 2004, 35, 223–240. (6) Vadivelan, V.; Kumar, K. V. J. Colloid Interface Sci. 2005, 286, 90–100. (7) Kalavathy, M. H.; Karthikeyan, T. S.; Rajgopal, L. R. J. Colloid Interface Sci. 2005, 292, 354–362. (8) Bouberka, Z.; Kacha, S.; Kameche, M.; Elmaleh, S.; Derriche, Z. J. Hazard. Mater. 2005, 119, 117–124. (9) Won, S. W.; Kim, H. J.; Choi, S. H.; Chung, B. W.; Kim, K. J.; Yun, Y. S. Chem. Eng. J. 2006, 121, 37–43. (10) Sismanoglu, T. Colloids Surf., A 2007, 297, 38–45. (11) Erdem, M.; Ozverdi, A. Sep. Purif. Technol. 2006, 51, 240–246. (12) Hameed, B. H.; Din, A. T. M.; Ahmad, A. L. J. Hazard. Mater. 2007, 141, 819–825. (13) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514–16525.
attempted to shed light on the possible theoretical origins of the first- and second-order rate equations for adsorption.
Materials and Methods Granular activated carbon (GAC) from (Calgon Carbon Corp.) was used as adsorbent with a mean size of 2.8 mm, an apparent density of 450 kg m-3, and a particle density of 650 kg m-3. The GAC was carefully rinsed with distilled water and dried at 103 °C overnight before use. Antibiotics used in this study include three kinds of representative β-lactam antibiotics: Penicillin G (PCG), Ampicillin (AMP), and Cephalosporin C (CPC). Sodium salt of PCG, sodium salt of AMP, and zinc salt of CPC were all purchased from Sigma-Aldrich Pte Ltd., Singapore. Batch adsorption tests were carried out in a 2 L Erlenmeyer flask on a shaker at 25 °C. GAC at the dosage of 1-10 g was added into 1 L of antibiotic solution with known initial concentrations (PCG, 10-1500 mg L-1; AMP, 10-1000 mg L-1; CPC, 10-100 mg L-1). Concentrations of antibiotics were determined by HPLC (PerkinElmer Series 200) with a UV detector at 220 nm. For this purpose, an anion exchange column (Thermo, MA), BioBasic AX 150 mm × 4.6 mm, 5 µm particle size was used to separate antibiotics in the solution. The mobile phase was 10mM ammonia acetate CH3COONH4: acetonitrile CH3CN (90:10 by volume), and the flow rate was 1 mL min-1. The column temperature was set to 35 °C, and the injection volume was 20 µL. In cases where the concentration was below the detection limit, 10 mL of sample was preconcentrated using 30 mg Waters Oasis HLB cartridges (Waters, MA). Before use, the cartridges were preconditioned with 5 mL of each ACN and water. A wash step with 1 mL of 5% ACN (v/v) was applied after the sample loading. The cartridge was air-dried for about 5 min under vacuum to remove excess water. The analytes retained were eluted with 1 mL of ACN. The final elution volume was approximately 1 mL. Through this way, the sample could be preconcentrated by 10 times.
Transformation of Langmuir Kinetics According to Langmuir,14 adsorption can be regarded as a reversible process between adsorbent and adsorbate (or solute): (14) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361–1403.
10.1021/la801839b CCC: $40.75 2008 American Chemical Society Published on Web 09/13/2008
11626 Langmuir, Vol. 24, No. 20, 2008 ka
A + B 798 AB
Liu and Shen
(1)
kd
in which A is the solute, B is the adsorption site on the adsorbent surface, AB is the complex formed, ka is the adsorption rate constant, and kd is the desorption rate constant. The proportion of the surface occupied by solute (θt) is defined as
θt )
qt C0 - Ct ) qmax qmaxX
(2)
in which qt and qmax are the adsorption capacity of adsorbent at any time and its maximum value, C0 and Ct are the respective concentrations of adsorbate in solution at time zero and time t, while X represents the dosage of adsorbent. Equation 2 can be rearranged to
Ct)C0 -qmaxXθt
(3)
It is assumed in eq 1 that the forward adsorption rate is firstorder with respect to Ct and (1 - θt), respectively, and the desorption follows the first-order regarding the sites occupied (θt). Thus, the adsorption and desorption rates in eq 1 can be written as
ra)kaCt(1 - θt)
(4)
rd)kdθt
(5)
and the overall rate equation for adsorption can be further expressed as follows:
dθt ) ra - rd ) kaCt(1 - θt) - kdθt dt
(6)
Equation 6 has been often referred to as the Langmuir kinetics.15,16 In fact, the Langmuir kinetics has been widely applied to describe adsorption at solid surface or at liquid interface.15-19 Inserting eq 3 into eq 6 yields
dθt ) kaqmaxXθt2 - (kaC0 + kaqmaxX + kd)θt + kaC0 (7) dt Equation 7 shows that the adsorption rate is a quadratic function of θt at given C0 and X. When adsorption reaches equilibrium, dθt/dt becomes zero, and θt reaches its value at equilibrium (θe). Solving eq 7 for the equilibrium coverage fraction (θe) gives
θe )
ka(qmaxX + C0) + kd - √∆ 2kaqmaxX
(8)
in which ∆ is the discriminant of eq 7 and given by
∆ ) k2a (C0 - qmaxX)2 + 2kakd(C0 + qmaxX) + k2d
(9)
Substitution of eq 8 into eq 7 yields
dθt ) k1(θe - θt) + k2(θe - θt)2 dt
(10)
in which (15) Novak, L. T.; Adriano, D. C. J. EnViron. Qual. 1975, 4, 261–266. (16) Kuan, W. H.; Lo, S. L.; Chang, C. M.; Wang, M. K. Chemosphere 2000, 41, 1741–1747. (17) Baret, J. F. J. Colloid Interface Sci. 1969, 30, 1–12. (18) Li, B.; Geeraerts, G.; Joos, P. Colloids Surf., A 1994, 88, 251–266. (19) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189–201.
k1 ) √∆
(11)
k2 ) kaqmaxX
(12)
Equation 10 represents a new transformation of the Langmuir kinetics and shows that the overall adsorption rate (dθt/dt) is the combination of the first-order term k1(θe - θt) and the secondorder term k2(θe - θt)2; that is, the Langmuir kinetics indeed represents a hybrid rate equation with a variable reaction order of 1-2. It appears that the relative weights of first-order and second-order terms govern the approximation of eq 10 to the first-order or second-order kinetics.
Sufficient and Necessary Conditions for Reduction of Langmuir Kinetics to First- and Second-Order Rate Equations First-Order Rate Equation for Adsorption. If k1 is much greater than k2(θe - θt), eq 10 reduces to eq 13, known as the first-order rate equation for adsorption:
dθt ≈ k1(θe - θt) dt
(13)
In fact, k1 . k2(θe - θt) is the sufficient and necessary condition for the Langmuir kinetics to be simplified to the first-order rate equation, and it can be further rearranged to
θt > > θ* ) θe -
k1 k2
(14)
In the adsorption study, θt is related to adsorption time (t), and thus the time corresponding to θ* is designated as t*. According to eq 14, two cases can be seen: (i) If k1/k2 is greater than θe, then θt > θ* is always valid as θt is positive. Thus, the Langmuir kinetics can be reasonably reduced to eq 13. (ii) In cases where k1/k2 is very close to θe, that is, θ* or t* is very close to zero, eq 14 reduces to θt > 0, indicating that the entire adsorption curve would be reasonably described by the first-order rate equation. Second-Order Rate Equation for Adsorption. If k1 , k2(θe - θt), eq 10 can be simplified to
dθt ≈ k1(θe - θt)2 dt
(15)
which is a typical second-order rate equation for adsorption. In fact, k1 , k2(θe - θt) is the sufficient and necessary condition for the Langmuir kinetics to be reduced to the second-order kinetics, and it can be further expressed as
θt < < θ* ) θe -
k1 k2
(16)
At k1/k2 , θe, θ* is very close to θe. In this case, adsorption kinetics can be reasonably described by a second-order rate equation (eq 15) over the whole adsorption period. Hybrid-Order Rate Equation for Adsorption. In addition to the above discussion, in cases where k1/k2 is neither close to zero nor close to θe, the Langmuir kinetics cannot be simplified to the first-order or second-order rate equations, and adsorption needs to be described by the Langmuir kinetics (eq 10) with a varying reaction order of 1-2. Calculation of θe and k1/k2. As discussed above, simplification of the Langmuir kinetics to first- or second-order rate equation
Langmuir Kinetics/Rate Equations for Adsorption
Langmuir, Vol. 24, No. 20, 2008 11627
Table 1. Values of KL and qmax, C0L, C0H, (k1/k2)min, (θe)critical, and (C0)critical for Antibiotics Adsorption by GAC at 25 °C (R: Correlation Coefficient) Langmuir isotherm
boundary conditions for simplification of Langmuir kinetics
antibiotics
qm (mg g-1)
KL (L mg-1)
R
C0L (mg L-1)
C0H (mg L-1)
(k1/k2)min
(θe)critical
(C0)critical (mg L-1)
PCG AMP CPC
427.3 164.2 33.67
0.02788 0.02261 0.4635
0.997 0.999 0.924
1144 518 89
4126 1446 328
0.26 0.46 0.23
0.87 0.77 0.89
2100 777 166
is governed by the relative magnitude of θe over k1/k2. At equilibrium, eq 2 becomes
θe )
C0 - Ce qmaxX
good the second-order rate equation is over the first-order rate equation or vice versa.3-12 Recently, it was shown that R may not be sensitive enough to distinguish goodness of curve fittings
(17)
which can be used to determine θe under given experimental conditions. According to eqs 9, 11, and 12, the k1/k2 can also be expressed as
k1 √k2a (C0 - qmaxX)2 + 2kakd(C0 + qmaxX) + k2d ) (18) k2 kaqmaxX When adsorption reaches its equilibrium, the following Langmuir isotherm can be derived from eq 6:
qe ) qmax
CeKL CeKL + 1
(19)
in which KL is the equilibrium constant of adsorption and is defined as:
KL )
ka kd
(20)
Substitution of eq 20 into eq 18 yields
k1 √KL2(C0 - qmaxX)2 + 2KL(C0 + qmaxX) + 1 ) k2 KLqmaxX
(21)
Under given conditions, C0 and X in eq 21 are known, and KL and qmax are obtainable from eq 19. Thus, eq 21 can be used for calculating the k1/k2 ratio.
Results The adsorption experiments of three model antibiotics were conducted at a fixed GAC dose of 5 g L-1, while C0 varied in a large concentration range. The equilibrium data were fitted to the Langmuir isotherm (eq 19), and qmax and KL were estimated using a nonlinear regression method provided by Matlab 7 (Table 1). Table 1 shows that the Langmuir isotherm can provide a satisfactory description of the data obtained, indicated by the high correlation coefficients of 0.924-0.997. It was found that GAC has a substantial adsorption capacity for these β-lactam antibiotics, and the adsorption capacity of antibiotics by GAC is in the order of PCG > AMP > CPC. The kinetic data of adsorption were fitted to the Langmuir, the first-order, and the second-order rate equations, respectively (Figure 1). It can be seen that the Langmuir kinetics can offer the best description of the data obtained at different initial antibiotic concentrations, while the goodness of the curving fitting by the first- and secondorder rate equations appears to be dependent on C0. In this study, θe and k1/k2 were calculated using eqs 17 and 21, and their values are summarized in Table 2. The adsorption data of antibiotics by GAC obtained at various C0 were fitted to the first-order, second-order, and Langmuir kinetics (eqs 10, 13, and 15). In a previous study of adsorption kinetics, the correlation coefficient (R) or R2 has been commonly used to compare how
Figure 1. Adsorption kinetics of the antibiotics by GAC at various C0. Solid line, prediction by Langmuir kinetics; dotted line, prediction by first-order kinetics; dashed line, prediction by second-order kinetics.
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Table 2. Predicted Adsorption Kinetics of Antibiotics by GAC no.
antibiotics
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
PCG
AMP
CPC
f value
C0 (mg L-1)
experimental θe by eq 17
theoretical θe by eq 24
theoretical k1/k2 by eq 21
theoretical θe - k1/k2
predicted kinetics
first-order
second-order
Langmuir
20 100 200 1000 1500 20 100 200 800 1000 10 20 40 60 80 100
0.0092 0.047 0.092 0.45 0.67 0.023 0.12 0.23 0.77 0.88 0.055 0.11 0.23 0.35 0.46 0.52
0.0092 0.046 0.092 0.45 0.67 0.023 0.12 0.23 0.78 0.87 0.059 0.12 0.23 0.35 0.46 0.58
1.01 0.97 0.93 0.58 0.38 1.03 0.95 0.84 0.47 0.54 0.96 0.90 0.78 0.67 0.56 0.45
