Article pubs.acs.org/jced
Modified Correlations for Adsorption Isotherms Fares D. Alsewailem* King Abdulaziz City for Science and Technology (Kacst), P.O. Box 6086, Riyadh 11442, Saudi Arabia ABSTRACT: In this research, new correlations for adsorption were developed. Classical adsorption isotherm models such as Freundlich, Langmuir, and Redlich−Peterson were modified to account for the effect of temperature by utilizing a superposition shift factor. The new correlations contain two independent variables; concentration (Ce) and temperature (T). The modified correlations developed by this research were validated by the results of three independent studies reported by others in the past, and were found to perfectly predict the experimental adsorption data presented by these three studies. Furthermore, by using these modified correlations one may get the amount of adsorbate per adsorbent (Qe) at any concentration and temperature by only knowing the adsorption isotherm data, that is, Qe vs Ce, at a single reference temperature. This is not possible with the use of the existing adsorption isotherm models since a change in temperature changes the parameters of the equation as well.
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INTRODUCTION The adsorption of gases and liquids on solid interfaces is an important process that is associated with many important applications in industry such as catalytic reactions, removal of contaminants from water, and capturing of CO2. Adsorption may be physical or chemical depending on the nature of both adsorbent and adsorbate. Several models were suggested in the past to describe the adsorption process; perhaps the most famous models were developed by Langmuir1 and Freundlich2 in early twentieth century. The Freundlich model is given by the following relation: Q e = k f Y (1/ n)
where kL and a are constants. The Freundlich and Langmuir models are called two-parameters models. Redlich and Peterson (R-P)3 suggested a three-parameter model to account for both homogeneous and heterogeneous adsorption isotherms that were best described by Freundlich and Langmuir, respectively. In other words, the R-P model was developed to combine advantages of both Freundlich and Langmuir models. The R-P model is given by the following relation: Q e = kRPCe/(1 + αCe β)
where kRP, α, and β are constants. One can realize the dependence nature of the R-P model given by eq 3 on Freundlich and Langmuir models given by eq 1 and eq 2; that is, when the exponent β approaches unity, the R-P model is reduced to Langmuir’s model. On the other hand, when the value of β is high, the R-P model matches Freundlich’s model. Despite the usefulness of the above-discussed models in describing various adsorption processes, these models may be used for determination of the relationship between Qe and Ce at isothermal conditions, and when the temperature changes a new set of parameters must be obtained, that is, kf, n, kL, a, kRP, α, and β, in order to accurately utilize these isotherm models. For this reason and as an attempt to overcome the shortcomings of the above models, a new form of adsorption isotherm model is developed by current study which incorporates two independent variables; Ce and temperature (T). This new model was developed by using a superposition shift factor.
(1)
where Y is pressure (P) for gas adsorption or concentration (C) for liquid adsorption, Qe = x/m, where x is the quantity adsorbed and m is the mass of the adsorbent, and kf and n are empirical constants for each adsorbent−adsorbate system at a certain temperature. Equation 1 has an asymptotic maximum as pressure or concentration increases. Higher values of pressure or concentration will lead to the saturation of the adsorbent surface. Mainly, the Freundlich model is good to describe adsorption of molecules on heterogeneous adsorbent surfaces. The Langmuir model was derived from kinetic studies based on the assumption that there exists an equivalent number of adsorption sites where only one molecule of adsorbate may be attached to the surface of the adsorbent. Bonding of a molecule to the adsorption site either chemically or physically should be strong to prevent desorption. The Langmuir model is based on the monomolecular adsorption on homogeneous adsorbent surfaces. It is given by the following relation: Q e = kLCe/(1 + aCe)
Received: September 24, 2014 Accepted: February 9, 2015
(2) © XXXX American Chemical Society
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DOI: 10.1021/je5008839 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
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THEORY The adsorption isotherm models given by eq 1,2, and 3 relate concentration at equilibrium (Ce) to amount of adsorbate per amount of adsorbent (Qe) at a constant temperature. Now define a shift factor, Sf = Qe(T)/Qe(Tref), as analogous to the shift factor used in rheology to estimate viscosity at a certain temperature with respect to a reference temperature. Qe may be related to temperature with the help of the Arrhenius relation as follows: Q e = A 0 e(E / RT )
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RESULTS AND DISCUSSION To use the modified correlations proposed by current study, that is, equations 9, 10, and 11, one needs to know E/R values and Qe vs Ce at a reference temperature. This may be done by plotting ln Qe versus 1/T as per eq 4 for the experimental data given in refs 4 to 6. After that, data at any temperature and concentration may be obtained by shifting the data of Qe vs Ce at the reference temperature. Representative examples of the calculation of E/R values for these previous studies are shown in Figures 1 and 2 and Tables 1 and 2. These previous studies
(4)
where A0 is the Arrhenius (pre-exponential) factor, R is the universal gas constant, and E is the activation energy. Here the power of the exponential may be positive if Qe decreases with increasing temperature (exothermic adsorption) and when Qe increases with increasing temperature which is in the case of endothermic adsorption, the exponent power will be negative. Now, one may write Qe at any temperature (T) and at a reference temperature (Tref) as Q e(T ) = A 0 e(E / RT )
(5)
Q e(Tref ) = A 0 e(E / RTref )
(6) Figure 1. Calculation of activation energies for the experimental data of a previous study reported in the literature.4
and hence the shift factor may be obtained as follows: Sf = Q e(T )/Q e(Tref ) = e E / R(1/ T − 1/ Tref )
(7)
Q e(T ) = Q e(Tref ) e E / R(1/ T − 1/ Tref )
(8)
or
Qe(Tref) is the adsorption isotherm relation given by eq 1, 2, and 3. Having said that, one may rewrite eq 8 in terms of temperature and concentration as follows: Modified Freundlich model (M-F): Q e(T , Ce) = k f Ce(1/ n) e E / R(1/ T − 1/ Tref )
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Modified Langmuir model (M-L): Q e(T , Ce) = kLCe/(1 + aCe) e E / R(1/ T − 1/ Tref )
(10)
Figure 2. Calculation of activation energies for the experimental data of a previous study reported in the literature.5 Reprinted with permission from ref 5. Copyright 2010 Elsevier.
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Table 1. Values of E/R Obtained from Figure 1
Modified Redlich-Peterson model (M-RP): Q e(T , Ce) = kRPCe/(1 + αCe β)e E / R(1/ T − 1/ Tref )
at T = Tref, eq 9 and 10 will be reduced to eq 1 and 3, that is, isothermal conditions. Here one may obtain Qe at any concentration and temperature provided that model parameters and activation energy (E) are known at Tref for a specific adsorbent/adsorbate system.
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VALIDATION OF THE MODELS Three independent studies which were reported in the past literature have been used to validate the models modified by current study. The first study is concerned with adsorption of lead ions on clay adsorbent where the adsorption capacity (Qe) increases with increasing temperature.4 Second and third studies reported the adsorption isotherm of phenol5 and toluene6 on activated carbon fibers (ACF) and activated carbon, respectively, where Qe decreases with increasing temperature.
Ce/ppm
(E/R)/K
R2
20 40 60 80 100
−296.8 −288.43 −281.78 −286.75 −285.85
0.9913 0.9987 0.9988 0.9994 0.9997
were chosen due to wide range of temperatures used to test adsorption isotherm and also they represent two different extremes of heat regimes, that is, endothermic and exothermic. Qe vs Ce at various temperatures was calculated using the parameters specified by these previous studies chosen to validate the models developed by current study.4−6 Figure 3 shows the Langmuir adsorption of lead ions on natural clay at temperatures ranging from 20 °C to 80 °C which was replotted from the first study.4 In the same graph, Figure 3, data at B
DOI: 10.1021/je5008839 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
Article
Table 2. Values of E/R Obtained from Figure 2 Ce/(mmol/L)
(E/R)/K
R2
0.1 0.2 0.3 0.4 0.5 0.6
1410 1084.4 893.57 758.25 653.44 567.37
0.9866 0.9726 0.9554 0.9344 0.9092 0.8783
represented by the second study.5 Here the adsorption data at the reference temperature, 25 °C, was shifted to get Qe (T, Ce) at various temperatures. Figure 4 also shows the prediction of the Freundlich adsorption of phenol on ACF at higher temperatures of 65 °C and 75 °C, which was not reported by the second study.5 The calculation of the adsorption at higher temperature beyond the range specified by the second study5 was done by applying the M-F model proposed by the current study. Figure 5 gives a comparison of the adsorption of phenol
Figure 5. Redlich−Peterson adsorption of phenol on activated carbon fibers. Comparison of experimental data replotted from literature5 with those predicted by modified Redlich−Peterson relation (M-RP), eq 11. Data at 65 °C and 75 °C was only calculated by eq 11.
Figure 3. Langmuir adsorption of lead ions on natural clay. Comparison of experimental data replotted from literature4 with those predicted by the modified Langmuir relation (M-L), eq 10.
on ACF using the Redlich−Peterson model calculated from the second study5 with the prediction of the same data using the M-RP model proposed by this research. One can clearly see that the M-RP model given by eq 11 can perfectly match the original experimental data reported by Liu et al.5 The prediction of phenol adsorption on ACF at higher temperatures was also shown in Figure 5 which was calculated by eq 11 utilizing the Qe vs Ce at the reference temperature reported by Liu et al.5 Lastly, Figure 6 shows the prediction of the
reference temperature of 20 °C was used to calculate Qe vs Ce at same temperatures used by the first study,4 40 °C, 60 °C, and 80 °C, but by using the modified Langmuir model (M-L) given by eq 10. It is seen from Figure 3 that the M-L correlation developed by current study perfectly predicts the behavior of lead adsorption on natural clay that was reported by the experimental data.4 By knowing only one set of Qe vs Ce at a single temperature, for example, Tref, other data of Qe vs Ce at any temperature may be obtained utilizing the superposition shift-factor relation given by eq 10. Figure 4 represents experimental data of the Freundlich adsorption of phenol on ACF at three different temperatures, 25 °C, 40 °C, and 55 °C, which were reported by others.5 It is also shown in Figure 4 that the prediction of adsorption of phenol on ACF using the M-F model given by eq 9 fairly matches the experimental data
Figure 6. Freundlich adsorption of toluene on activated carbon. Comparison of experimental data replotted from literature6 with those predicted by modified Freundlich relation (M-F), eq 9.
experimental data reported by the third study for the adsorption of toluene on activated carbon by applying eq 9, that is,M-F. One can see from Figure 6 that M-F fairly matches the experimental data reported by Benkhedda et al.6 at a temperature above the reference temperature.
Figure 4. Freundlich adsorption of phenol on activated carbon fibers. Comparison of experimental data replotted from literature5 with those predicted by modified Freundlich relation (M-F), eq 9. Data at 65 °C and 75 °C was only calculated by eq 9. C
DOI: 10.1021/je5008839 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
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Article
CONCLUSIONS New correlations based on basic adsorption isotherm models such as Freundlich’s and Langmuir’s were developed by this study utilizing the principle of superposition shift factor. Contrary to conventional adsorption isotherm models, these new correlations have two independent variables; concentration and temperature. For an adsorbent−adsorbate system, one may predict Qe vs Ce by using the modified adsorption correlation given by the current study at any temperature by only knowing the adsorption data at a single reference temperature and the activation energy of such a system.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Funding
The author thanks his employer, Kacst, for its continuous support. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Langmuir, I. The adsorption of gases on plane surfaces of glass, mica and platinum. J. Am. Chem. Soc. 1918, 40, 1361−1402. (2) Freundlich, H. Colloid and Capillary Chemistry; Methuen: London, 1926. (3) Redlich, O.; Peterson, D. L. A useful adsorption isotherm. J. Phys. Chem. 1959, 63, 1024−1026. (4) Aljlil, S. A.; Alsewailem, F. D. Lead uptake by natural clay. J. Appl. Sci. 2009, 9, 4026−4031. (5) Liu, Q.-S.; Zheng, T.; Wang, P.; Jiang, J.-P.; Li, N. Desorption isotherm, kinetic and mechanism studies of some substituted phenol on activated carbon fibers. Chem. Eng. J. 2010, 157, 348−356. (6) Benkhedda, J.; Jaubert, J.-N.; Barth, D.; Perrin, L. Experimental and modeled results describing the adsorption of toluene onto activated carbon. J. Chem. Eng. Data 2000, 45, 650−653.
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DOI: 10.1021/je5008839 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
