Research Note Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
pubs.acs.org/IECR
Erroneous Application of Pseudo-Second-Order Adsorption Kinetics Model: Ignored Assumptions and Spurious Correlations Ye Xiao, Jalel Azaiez, and Josephine M. Hill* Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Dr NW, Calgary, AB Canada, T2N 1N4 ABSTRACT: In this research note, we revisit the pseudo-second-order model for adsorption kinetics, its assumptions, and its application to simulated, random, and published data. In particular, a widely used linear form of the pseudo-second-order modelplotting t/qt against tis shown to result in spurious correlations for typical adsorption experimental data. Depending on the range of data used, data from pseudo-first-order and pseudo-third-order models can also appear to be well-fit by the pseudo-second-order model. Inspection of the residual errors, however, indicates that the errors are not randomly distributed, as they should be. Based on this study, it is recommended to always verify the assumptions of a model, fit the data with the nonlinear form of the model equation, and inspect the residual plot to determine the goodness of fit.
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INTRODUCTION
RESULTS AND DISCUSSION Pseudo-Second-Order Model Development. If the adsorption rate is pseudo-second-order, with respect to the adsorption sites, the kinetics equation can be expressed as eq 1.2,16 In general, the term “pseudo” is associated with a kinetics model when the concentration of one reactant is essentially constant (i.e., fed in excess), so that it appears to have no effect on the reaction rate.17 However, here, in adsorption kinetics, it was called “pseudo” because the equation was based on the amount adsorbed on the adsorbent instead of the concentration of reactants in the liquid.15
Adsorption is an effective and efficient method to remove hazardous substances from water. To put the performance of an adsorbent in context, adsorption isotherms and adsorption kinetics are collected. A model is fit to these data for quantification and process design. Many models have been developed to describe the kinetics of adsorption based on reaction control (e.g., pseudo-first-order1 and pseudo-secondorder2 models), or diffusion control3−5(e.g., the Weber−Morris model). One of the most commonly applied models is the pseudo-second-order model, which was developed by Ho and Mckay in 1998.2 Despite being developed on the basis of a reaction-controlled adsorption, the pseudo-second-order model has been widely applied to various adsorbents, including granular activated carbon,6−8 for which diffusion control would be expected. In addition, it has also been widely used for different adsorbates, such as single-charged ammonium9,10 and fluoride,11,12 for which a pseudo-second-order mechanism is highly unlikely. Researchers often ignore the assumptions of the model and base the applicability of the model solely on the determination coefficients (R2) obtained. Application of a linear form of the pseudo-second-order model, in which t/qt is plotted vs t,13−15 often results in very high correlation coefficients (R2 > 0.99), regardless of the actual model fit. The fact that so many adsorption processes with widely different adsorbents and adsorbates were well fit by this one model prompted us to investigate further. In this study, we revisited the assumptions of the pseudo-second-order adsorption kinetics model, analyzed the determination coefficients obtained with various sets of data, and analyzed the applicability of the model to several sets of literature data. © XXXX American Chemical Society
dqt dt
= k(qe − qt )2
(1)
where qt (mg/g) is the amount adsorbed at time t (min), qe (mg/g) the equilibrium adsorption capacity, and k (g/mg min) the adsorption rate constant. The adsorption reaction corresponding to eq 1 may be written as follows:16,18 k1
A + 2S XooY AS2 k −1
(2)
where A is the adsorbate, S a vacant adsorption site, and AS2 the adsorbate adsorbed on two sites. With the assumption that the adsorption process is not limited by diffusion and eq 2 is an elementary step, the reaction rate is expressed as Received: Revised: Accepted: Published: A
November 15, 2017 January 8, 2018 February 6, 2018 February 6, 2018 DOI: 10.1021/acs.iecr.7b04724 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Research Note
Industrial & Engineering Chemistry Research
Figure 1. Adsorption kinetics for (a) phosphate adsorption on unrinsed tamarind nut shell activated carbon,21 and (b) Methyl Orange adsorption on granular activated carbon.6 Solid lines are nonlinear fits to the pseudo-second-order model.
Figure 2. Impact of different variables on the coefficient of determination (R2): (a) as a function of the correlation coefficient (rt,qt) for different relative values of t and qt, and (b) demonstration of the fit of a random set of data to eq 7.
Figure 3. (a) Simulated adsorption kinetics data for pseudo-first-, pseudo-second-, and pseudo-third-order data with rate constants of 0.01 min−1, 0.001 L mg−1 min−1, and 0.0001 L2 mg−2 min−1, respectively. (b) Simulated fit to a linear form of the pseudo-second-order model (eq 7), where the data was generated from first-order (○), second-order (●), and third-order (△) models. The maximum adsorption capacity was set to be the same for the three models.
B
DOI: 10.1021/acs.iecr.7b04724 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Research Note
Industrial & Engineering Chemistry Research dC AS2 dt
= k1CACS2 − k −1C AS2
(3)
where Ci corresponds to the concentration of species i at adsorption time of t, k1 the rate constant for the forward reaction, and k−1 the rate constant for the reverse reaction. In order to simplify eq 3 to eq 1, two other assumptions must be made: first, the adsorption is irreversible so that the rate of desorption, k−1CAS2, can be neglected, and second, the concentration of adsorbate in the liquid remains essentially constant during the adsorption process (i.e., CA is a constant).19 The second assumption is rarely valid in studies reported in the literature. With the addition of the following relationships to relate the concentrations of adsorbed species and vacant sites to the amounts adsorbed at time t and at equilibrium, in eqs 4 and 5, respectively, eq 1 is obtained from eq 3. Figure 4. Residual plots for data generated from pseudo-first- (○), pseudo-second- (●), and pseudo-third-order (△) adsorption kinetics models fit nonlinearly to the pseudo-second-order model (eq 6).
C AS2 ∝ qt
(4)
CS ∝ (qe − qt )
(5)
According to Azizian, the pseudo-second-order model expression (eq 1) could be obtained if the adsorption reaction follows a first-order mechanism (with respect to the adsorption site) but the adsorbate concentration in the liquid phase changes significantly during adsorption and negligible desorption.20 Thus, an R2 value of ∼1 would be misleading in this situation. To further illustrate the importance of verifying that the model assumptions are valid, several sets of literature data were analyzed. Bhargava and Sheldarkar used an unrinsed tamarind nut shell activated carbon to adsorb phosphate from water, with the results plotted in Figure 1a.21 A nonlinear form of the pseudo-second-order adsorption model fit the data with an R2 value of 0.9459 (using SigmaPlot 13.0). A linear form of this model was used and an R2 value of 0.998 was obtained,2 but the authors stated that Zn ions had leached from the activated carbon, resulting in the removal of phosphate mainly via the precipitation of zinc phosphate and not a second-order adsorption process.21 In a second example, León et al. investigated the adsorption of Methyl Orange on granular activated carbon from water.6 As plotted in Figure 1b, the value
Figure 5. Residual plot for data generated from pseudo-first-order kinetics model with different rate constants ((▽) k = 0.005 min−1; (□) k = 0.0005 min−1) and (■) the Weber−Morris model fit nonlinearly to the pseudo-second-order model (eq 6).
Figure 6. Adsorption kinetics of Victoria blue dye from aqueous solutions by fly ash at different initial concentrations ((●) 1.0 × 10−4 M, (○) 4.5 × 10−4 M, and (▲) 8.5 × 10−4 M): (a) nonlinear fitting to pseudo-second-order model (eq 6), and (b) residual plots. Data taken from Khare et al.30 C
DOI: 10.1021/acs.iecr.7b04724 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Research Note
Industrial & Engineering Chemistry Research of R2 was 0.9926 for the pseudo-second-order model fit. Given the large particle size (2−5 mm) and microporosity of the adsorbent,6 and the low diffusion coefficient of Methyl Orange (∼6.0 × 10−6 cm2/s),22 the adsorption process was likely diffusion-limited, thus violating one of the assumptions of the pseudo-second-order model. Linearization of the Model and Subsequent Model Fitting. Using the initial condition of qt = 0 at t = 0, the integral form of eq 1 is as follows: qt =
capacity. The curve shapes in Figure 2a vary, depending on the relationship between Ct and Cqt. In an actual experiment, this relationship will change over time. That is, initially qt increases rapidly (Ct < Cqt) while at longer times, qt is relatively constant (Ct ≫ Cqt). Figure 2b illustrates the fit obtained using eq 7 (i.e., t/qt vs t, red points) with data randomly distributed between 0 and 1000 min for t and 100−150 mg/g for qt. The value of R2 was 0.9463, despite there being no correlation between the data. In addition, if the relationship between qt and t is linear (rqt,t = 1), the R2 value will be 1.0 for eq 7. To further illustrate the problems with fitting data using eq 7, two sets of simulated data were generated, using pseudo-firstorder and third-order adsorption kinetics models, and these data are shown in Figure 3a. The curves are distinct initially but overlap at longer times. By using the linear form of eq 7 to fit the simulated data, R2 values of >0.99 were obtained (Figure 3b) for both sets of data. Relying only on the coefficient of determination, the model fits would incorrectly lead to the conclusion that the adsorption processes all followed a pseudosecond-order mechanism. Recommendations for Fitting Adsorption Data. As has been suggested,23,28,29 a nonlinear fitting method is superior to a linear fitting method for the pseudo-second-order model. Using a nonlinear regression (SigmaPlot software) with the pseudo-first- and pseudo-third-order model data in Figure 3a, however, resulted in R2 values of 0.973 and 0.952, respectively. The values of R2 are not always reported but values of >0.95 are generally accepted as a “good” fit. Thus, an inappropriate conclusion could still be drawn and additional assessments of the fit are required. A residual plot is one type of assessment. The standardized residuals are calculated by using the following equation:
qe 2kt 1 + qekt
(6)
In order to obtain the adsorption kinetics parameters, a linear form of eq 6 has often been used. There are many linear forms,23 but eq 7 has been most widely used,7−15,24−26 possibly because this form was proposed when Ho and McKay developed the pseudo-second-order adsorption kinetics model.2 1 1 t = + t qt qe kqe 2
(7)
In a typical adsorption experiment, the adsorbed amount increases quickly initially as the most vacant sites are available. As adsorption progresses, the rate of adsorption decreases as the equilibrium adsorption capacity is approached. If data are collected at longer times, the variation of adsorption time (t) is much bigger than the variation of adsorption capacity (qt) with the result that the fit to eq 7 improves. According to Chayes, R2 for the fit of eq 7 to data can be estimated using eqs 8−12.27 R t /q ,t 2 = t
(Ct − rq , tCq )2 t
t
Ct 2 + Cq 2 − 2rq , tCtCq t
t
t
n
1 x̅ = n
∑ xi
sx =
1 n−1
standardized residual = (9)
i=1
∑ (xi − x ̅ )2 (10)
i=1
1 n−1
(11) n
∑i = 1 (xi − x ̅ )(yi − y ̅ ) sxsy
qt − qt′ s
(13)
where q′t (mg/g) is the adsorbed amount at time t predicted by the model, and s is the standard deviation of the residuals (mg/ g). If a model, such as the pseudo-second-order model, truly represents the behavior of the data, the residuals should be randomly distributed and close to zero. The residuals calculated for the nonlinear fitting of the data in Figure 3a using eq 6 are shown in Figure 4. In this plot, the poor fits of the pseudosecond-order model to the data generated from the pseudofirst- and pseudo-third-order equations are clear: the residuals are not randomly distributed around zero. As further examples, R2 values of 1.000 were obtained for the fits of simulated pseudo-first-order data with rate constants of either 0.005 min−1 or 0.0005 min−1, while an R2 value of 0.981 was obtained for simulated data from a Weber−Morris model.3 Again, spurious conclusions may be drawn based only on the R2 values. However, the residual plots (Figure 5) clearly show that the standardized residuals are not randomly distributed, and the pseudo-second-order model is unsuitable for all three sets of data. Finally, data from the literature was examined. Khare et al.30 investigated the adsorption of Victoria blue (VB) dye ([C33H40N3]OH) by fly ash at different initial dye concentrations and their results are plotted in Figure 6a. Using a nonlinear fitting method, the R2 values for the pseudo-secondorder model fit are 0.934, 0.995, and 0.995 for initial VB concentrations of 1.0 × 10−4 M, 4.5 × 10−4 M, and 8.5 × 10−4
n
s Cx = x x̅ rxy =
(8)
(12)
2
In these equations, Rt/qt,t is the determination coefficient calculated for eq 7; Ct and Cqt are the variation coefficients of t and qt respectively, which are calculated by using eq 11; and rqt,t is the correlation coefficient between t and qt which is calculated according to eq 12. To illustrate the relationships between these parameters, several sets of simulated data were generated by changing the relative values of the variation coefficients, Ct and Cqt. As shown in Figure 2a, the value of Rt/qt,t2 is dependent on both the correlation between t and qt (i.e., rt,qt) and the relative values of the variation coefficients. For experiments over longer times (i.e., larger Ct values, relative to Cqt), the value of R2 is ∼1 regardless of the correlation between t and qt. Figure 2a illustrates how the fit of eq 6 to even random data (rt,qt = 0) can result in high (>0.8) values of R2 when the change in time is large, compared to the change in adsorption D
DOI: 10.1021/acs.iecr.7b04724 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Research Note
Industrial & Engineering Chemistry Research
(9) Huang, H.; Xiao, X.; Yan, B.; Yang, L. Ammonium removal from aqueous solutions by using natural Chinese (Chende) zeolite as adsorbent. J. Hazard. Mater. 2010, 175 (1−3), 247−252. (10) Wahab, M. A.; Jellali, S.; Jedidi, N. Ammonium biosorption onto sawdust: FTIR analysis, kinetics and adsorption isotherms modeling. Bioresour. Technol. 2010, 101 (14), 5070−5075. (11) Fan, X.; Parker, D. J.; Smith, M. D. Adsorption kinetics of fluoride on low cost materials. Water Res. 2003, 37 (20), 4929−4937. (12) Ma, W.; Ya, F.-Q.; Han, M.; Wang, R. Characteristics of equilibrium, kinetics studies for adsorption of fluoride on magneticchitosan particle. J. Hazard. Mater. 2007, 143 (1−2), 296−302. (13) Gujar, R. B.; Ansari, S. A.; Mohapatra, P. K. Highly Efficient Composite Polysulfone Beads Containing a Calix[4]arene−Monocrown-6 Ligand and a Room Temperature Ionic Liquid for Radiocesium Separations: Remediation of Environmental Samples. Ind. Eng. Chem. Res. 2016, 55 (48), 12460−12466. (14) Zhang, M.; Gao, Q.; Yang, C.; Pang, L.; Wang, H.; Li, H.; Li, R.; Xu, L.; Xing, Z.; Hu, J.; Wu, G. Preparation of Amidoxime-Based Nylon-66 Fibers for Removing Uranium from Low-Concentration Aqueous Solutions and Simulated Nuclear Industry Effluents. Ind. Eng. Chem. Res. 2016, 55 (40), 10523−10532. (15) Ho, Y.-S. Review of second-order models for adsorption systems. J. Hazard. Mater. 2006, 136 (3), 681−689. (16) Ho, Y. S.; McKay, G. The kinetics of sorption of divalent metal ions onto sphagnum moss peat. Water Res. 2000, 34 (3), 735−742. (17) Laidler, K. J. A glossary of terms used in chemical kinetics, including reaction dynamics (IUPAC Recommendations 1996). Pure Appl. Chem. 1996, 68 (1), 149−192. (18) Ho, Y. S.; McKay, G. Pseudo-second order model for sorption processes. Process Biochem. 1999, 34 (5), 451−465. (19) Blanchard, G.; Maunaye, M.; Martin, G. Removal of heavy metals from waters by means of natural zeolites. Water Res. 1984, 18 (12), 1501−1507. (20) Azizian, S. Kinetic models of sorption: a theoretical analysis. J. Colloid Interface Sci. 2004, 276 (1), 47−52. (21) Bhargava, D. S.; Sheldarkar, S. B. Use of TNSAC in phosphate adsorption studies and relationships. Literature, experimental methodology, justification and effects of process variables. Water Res. 1993, 27 (2), 303−312. (22) Mitsuishi, M.; Datyner, A. Diffusion of methyl orange and its homologs in water and in micellar solution of dodecyltrimethylammonium bromide. Sen'i Gakkaishi 1980, 36 (4), T175−T178. (23) El-Khaiary, M. I.; Malash, G. F.; Ho, Y.-S. On the use of linearized pseudo-second-order kinetic equations for modeling adsorption systems. Desalination 2010, 257 (1−3), 93−101. (24) Chaudhuri, H.; Dash, S.; Sarkar, A. Fabrication of New Synthetic Routes for Functionalized Si-MCM-41 Materials as Effective Adsorbents for Water Remediation. Ind. Eng. Chem. Res. 2016, 55 (38), 10084−10094. (25) Yousef, R. I.; El-Eswed, B.; Al-Muhtaseb, A. a. H. Adsorption characteristics of natural zeolites as solid adsorbents for phenol removal from aqueous solutions: Kinetics, mechanism, and thermodynamics studies. Chem. Eng. J. 2011, 171 (3), 1143−1149. (26) Hui, K. S.; Chao, C. Y. H.; Kot, S. C. Removal of mixed heavy metal ions in wastewater by zeolite 4A and residual products from recycled coal fly ash. J. Hazard. Mater. 2005, 127 (1−3), 89−101. (27) Chayes, F. On ratio correlation in petrography. J. Geol. 1949, 57, 239−254. (28) Lin, J.; Wang, L. Comparison between linear and non-linear forms of pseudo-first-order and pseudo-second-order adsorption kinetic models for the removal of methylene blue by activated carbon. Front. Environ. Sci. Eng. China 2009, 3 (3), 320−324. (29) Ho, Y.-S. Second-order kinetic model for the sorption of cadmium onto tree fern: A comparison of linear and non-linear methods. Water Res. 2006, 40 (1), 119−125. (30) Khare, S. K.; Panday, K. K.; Srivastava, R. M.; Singh, V. N. Removal of victoria blue from aqueous solution by fly ash. J. Chem. Technol. Biotechnol. 1987, 38 (2), 99−104.
M, respectively. The residuals for these plots are given in Figure 6b. The residuals are not randomly distributed around zero and so the pseudo-second-order model is not a good fit to the data. The authors concluded that the adsorption of the dye was diffusion-limited, consistent with the pseudo-second-order model not being an appropriate model for these data.30
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CONCLUSIONS In this paper, the pseudo-second-order adsorption kinetics model is developed with the assumptions explicitly stated. These assumptions, which include (i) reaction-controlled adsorption, (ii) essentially constant liquid concentration, and (iii) no desorption, must be satisfied before the model is applied. If the assumptions are not satisfied, the coefficient of determination value (R2) will be meaningless, and a high value may be misinterpreted. Similarly, a spuriously high value of R2 will be obtained if a linear form of the pseudo-second-order model is used (i.e., plotting t/qt vs t). The use of a nonlinear form of the equation for the fitting should be followed by a check of the residual plot, and the model only said to be consistent with the data if the residuals are randomly distributed and close to zero. These issues have been illustrated with simulated and experimental data.
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AUTHOR INFORMATION
Corresponding Author
*Tel.: +1 403 210 9488. Fax: 1 403 284 4852. E-mail: jhill@ ucalgary.ca. ORCID
Ye Xiao: 0000-0003-4235-2154 Josephine M. Hill: 0000-0003-2708-565X Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors gratefully acknowledge funding from the Department of Chemical & Petroleum Engineering, University of Calgary for scholarships (Y.X.).
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REFERENCES
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DOI: 10.1021/acs.iecr.7b04724 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
