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Analysis of Kinetic Langmuir Model. Part I: Integrated Kinetic Langmuir Equation (IKL): A New Complete Analytical Solution of the Langmuir Rate Equation Adam W. Marczewski* Department of Radiochemistry and Colloid Chemistry, Faculty of Chemistry, Maria Curie-Skz odowska University, M. Curie-Skz odowska Sq. 3, 20-031 Lublin, Poland Received March 11, 2010. Revised Manuscript Received August 12, 2010 In the article, a new integrated kinetic Langmuir equation (IKL) is derived. The IKL equation is a simple and easy to analyze but complete analytical solution of the kinetic Langmuir model. The IKL is compared with the nth-order, mixed 1,2-order, and multiexponential kinetic equations. The impact of both equilibrium coverage θeq and relative equilibrium uptake ueq on kinetics is explained. A newly introduced Langmuir batch equilibrium factor feq that is the product of both parameters θequeq is used to determine the general kinetic behavior. The analysis of the IKL equation allows us to understand fully the Langmuir kinetics and explains its relation with respect to the empirical pseudo-first-order (PFO, i.e., Lagergren), pseudo-second-order (PSO), and mixed 1,2-order kinetic equations, and it shows the conditions of their possible application based on the Langmuir model. The dependence of the initial adsorption rate on the system properties is analyzed and compared to the earlier published approximate equations.
Introduction The adsorption process1 is one of the most common phenomena occurring in ecosystems, and it is an important tool in environmental protection and in many industrial processes. The adsorption of small toxic molecules and ions on natural sorbents2 present in soil and water sediments may limit their spreading and reduce their harmful effects far from their source. However, the long-term effects related to their delayed release may prove even more dangerous. Although adsorption (or generally sorption) effects are only one of many factors, their importance stems not only from adsorption equilibria but also in large part from adsorption and desorption kinetics. Adsorption kinetics depends on many factors; however, in most systems several very simple equations are used with success.3 The first successful kinetic equation was the pseudo-first-order Lagergren equation that well describes any system where the rate is proportional to the system’s distance from equilibrium.4,5 The second simple rate concept that is still used appeared in the derivation of the Langmuir isotherm6,7 and took into account the limited capacity of the adsorbent surface as well as both adsorption and desorption processes.8,9 The same idea is now used in the *Phone: þ48-81-5375624. Fax: þ48-81-5332811. Home page: www.guide. adsorption.org. E-mail:
[email protected]. (1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Deryzo-Marczewska, A.; Jaroniec, M. In Surface and Colloid Science; Matijevic, E., Ed.; Plenum Press: New York, 1987; Vol. 14, p 301. (3) Pzazinski, W.; Rudzinski, W.; Pzazinska, A. Adv. Colloid Interface Sci. 2009, 152, 2–13. (4) Lagergren, S. K. Sven. Vetenskapsakad. Handl. 1898, 24, 1–39. (5) Ho, Y. S. Scientometrics 2004, 59, 171–177. (6) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361–1403. (7) Langmuir, I. Phys. Rev. 1918, 8, 149–176. (8) Liu, Y.; Shen, L. Langmuir 2008, 24, 11625–11630. (9) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47–52. (10) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95–101. (11) Clark, C. A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (12) King, D. A.; Wells, M. G. Surf. Sci. 1972, 29, 454–482. (13) King, D. A. Surf. Sci. 1977, 64, 43–51.
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theory of activated/adsorption desorption (TAAD),10-15 which a modern version of the absolute rate theory (ART).16-20 Other important simple equations that appeared later described diffusion in porous solids21 and chemisorption.22 nth-order equations23-28 take into account multisite occupancy29 and strongly varying conditions during the adsorption process30,31 or in modified forms obtained by using fractal-like kinetics may describe even disordered and porous media.32 However, such purely nthorder equations do not contain separate adsorption and desorption terms, though both terms may be folded into a common one. Other equations try to mimic the behavior of experimental systems, where the rate seem to follow various equations within limited time or adsorption ranges (e.g., multiexponential equation,33-38 MPFO equation,39 and multistep equations35,40-43). (14) Talbot, J.; Jin, X.; Wang, N.-H. Langmuir 1994, 10, 1663–1666. (15) Rudzinski, W. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Surfaces; Rudzinski, W., Steele, W., Zgrablich, G., Eds.; Elsevier: New York, 1997; Chapter 6. (16) Arrhenius, S. Z. Phys. Chem. 1889, 4, 226. (17) Polanyi, M.; Wigner, E. Z. Phys. Chem. 1928, 139A, 439. (18) Eyring, H. J. Chem. Phys. 1935, 3, 107–115. (19) Eyring, H. Chem. Rev. 1935, 17, 65–77. (20) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, 1986. (21) Crank, J. Mathematics of Diffusion; Clarendon Press: Oxford, U.K., 1975. (22) Roginsky, S.; Zeldovich, Y. Acta Physicochim. USSR 1934, 1, 554. (23) Sobkowski, J.; Czerwinski, A. J. Electroanal. Chem. 1974, 55, 391–397. (24) Ho, Y. S.; McKay, G. Chem. Eng. J. 1998, 70, 115–124. (25) Ho, Y. S.; McKay, G. Process Safety Environ. Prot. 1998, 76B, 183–191. (26) Ho, Y. S. Water Res. 2006, 40, 119–125. (27) Cheung, C. W.; Porter, J. F.; McKay, G. Water Res. 2001, 35, 605–612. € (28) Ozer, A. J. Hazard. Mat. 2007, 141, 753–761. (29) Ritchie, A. G. J. Chem. Soc., Faraday Trans. 1 1977, 73, 1650–1653. (30) Blanchard, G.; Maunaye, M.; Martin, G. Water Res. 1984, 18, 1501–1507. (31) Ho, Y. S. J. Hazard. Mater. 2006, B136, 681–689. (32) Brouers, F.; Sotolongo-Costa, O. Physica A 2006, 368, 165–175. (33) Bonifazi, M.; Pant, B. C.; Langford, C. H. Environ. Technol. 1996, 17, 885–890. (34) Fletcher, A. S.; Cussen, E. J.; Bradshaw, D.; Rosseinsky, M. J.; Thomas, K. M. J. Am. Chem. Soc. 2004, 126, 9750–9759. (35) Marczewski, A. W. Appl. Surf. Sci. 2007, 253, 5818–5826. (36) Marczewski, A. W. Pol. J. Chem. 2008, 82, 271–281. (37) Marczewski, A. W.; Deryzo-Marczewska, A.; Skrzypek, I.; Pikus, S.; Kozak, M. Adsorption 2009, 15, 300–305.
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DOI: 10.1021/la1010049
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Currently, many rate equations are also developed within the framework of the statistical rate theory (SRT),44,45 especially for adsorption on energetically heterogeneous solids.3,46,47 The Erofeev-Kolmogorov equation48 is derived within the statistical nucleation and growth KEKAM theory49,50 and may be applied both to the isothermal and nonisothermal kinetic processes. This equation takes the form of a stretched exponential and is also generalized as the multiple stretched exponential equation.34 Most of the important adsorption systems are usually too complicated to be fully modeled and good adsorption data is difficult to obtain, and thus quite often simple equations are used in descriptions of equilibrium and kinetic adsorption data. Common problems related to the verification of the theoretical models are the experimental scatter as well as the typically toonarrow range of experimental data. Quite often special fitting methods such as linear PSOE plots may influence judgment if used improperly. To be able to understand more complicated systems better, well-defined model solids are now used in kinetic experiments. Mesoporous carbons and silicas are well suited for such model experiments; they may be obtained in controlled syntheses, allowing high adsorption capacities and the required pore size to be attained.51-55 The rates of adsorption in aqueous solutions are such that smooth kinetic curves may be obtained over the entire adsorption progress range.33,36-38 The complicated nature of such systems often makes it impossible to describe their kinetics by using only the fully theoretically justified equations. In such a case, we may use various empirical equations that are able to follow typical kinetic behaviors with contributions related to parallel or follow-up processes (e.g., a multiexponential equation (m-exp),38,56,57 multiple-step models,40,58-60 and the sum of stretched exponentials34). Such equations make it possible to include highly diverse rate coefficients related to the adsorbent particle external surface and the porous inside61 and the adsorbent energetic or structural heterogeneity62 and (38) Marczewski, A. W. Appl. Surf. Sci. 2010, 256, 5145–5152. (39) Yang, X. Y.; Al-Duri, B. J. Colloid Interface Sci. 2005, 287, 25–34. (40) Nagaoka, H.; Imae, T. J. Colloid Interface Sci. 2003, 264, 335–342. (41) Sarkar, D.; Chattoraj, D. K. J. Colloid Interface Sci. 1993, 157, 219–226. (42) Choi, J. W.; Choi, N.Ch.; Lee, S. J.; Kim, D. J. J. Colloid Interface Sci. 2007, 314, 367–372. (43) Nguyen, C.; Do, D. D. Langmuir 2000, 16, 1868–1873. (44) Rudzinski, W.; Pzazinski, W. J. Phys. Chem. B 2006, 110, 16514–16525. (45) Ward, C. A. J. Chem. Phys. 1977, 67, 229–235. (46) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599–5605. (47) Rudzinski, W.; Pzazinski, W. Appl. Surf. Sci. 2007, 253, 5827–5840. (48) Erofeev, B. V. Dokl. Akad. Nauk SSSR 1946, 52, 515–523. (49) Avrami, M. J. Chem. Phys. 1939, 7, 1103. Avrami, M. J. Chem. Phys. 1940, 8, 212. Avrami, M. J. Chem. Phys. 1941, 9, 177–184. (50) Gorbachev, V. M. J. Therm. Anal. 1978, 13, 509–514. (51) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710–712. (52) Zhao, D.; Huo, Q.; Feng, J.; Chmelka, B. F.; Stucky, G. D. J. Am. Chem. Soc. 1998, 120, 6024–6036. (53) Joo, S. H.; Jun, S.; Ryoo, R. Microporous Mesoporous Mater. 2001, 44/45, 153–158. (54) Kim, J.; Lee, J.; Hyeon, T. Carbon 2004, 42, 2711–2719. (55) Deryzo-Marczewska, A.; Marczewski, A. W.; Skrzypek, I.; Pikus, S.; Kozak, M. Appl. Surf. Sci. 2005, 252, 625–632. (56) Derylo-Marczewska, A.; Marczewski, A. W.; Winter, Sz.; Sternik, D. Appl. Surf. Sci. 2010, 256, 5164–5170. (57) Marczewski, A. W. Proceedings of the 11th Ukrainian-Polish Symposium on Theoretical and Experimental Studies of Interfacial Phenomena and Their Technological Applications; Krasnobrod-Zamosc, Poland, Aug 22-26, 2007, UMCS Press: Lublin; p 95, ISBN978-83-227-2717-1. (58) Wilczak, A.; Keinath, T. M. Water Environ Res. 1993, 65, 238–244. (59) Chiron, N.; Guilet, R.; Deydier, E. Water Res. 2003, 37, 3079–3086. (60) Li, Q.; Yue, Q. Y.; Su, Y.; Gao, B. Y.; Li, J. J. Hazard. Mater. 2009, 165, 1170–1178. (61) Brandt, A.; B€ulow, M.; Deryzo-Marczewska, A.; Goworek, J.; Schmeisser, J.; Sch€ops, W.; Unger, B. Adsorption 2007, 13, 267–279. (62) Marczewski, A. W. Proceedings of the 6th ISSHAC Symposium; Zakopane, Poland, Aug 28-Sept 02, 2006; UMCS Press: Lublin; p 263-264, ISBN 83-2272750-1.
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for dissociating adsorbates.63 By analysis of their parameter spectra,33,36-38,56,57,61-63 it is possible to assess the impact of various contributions. (For details, see Appendixes A and B available as Supporting Information for this article.) However, by no means does such an approach makes even simple theoretical models obsolete; rather, we are looking for ways to extend simple but fundamental models such as that of Langmuir6-9 to more complicated systems. We are also looking for reasons that some simple equations, such as nth-order empirical equations and their generalizations, are often very successful in their analysis of the kinetic data of many complicated systems.
Azizian’s Solution of the Langmuir Rate Equation A few years ago, Azizian showed that both first- and secondorder equations may be treated as special cases of a kinetic model leading to or being equivalent to the Langmuir isotherm.9 The kinetic Langmuir model6,7 assumes that the overall adsorption rate dθ/dt is a difference in adsorption (va) and desorption (vd) rates and that at equilibrium both rates are equal. The adsorption rate is proportional to the concentration (c) and the available adsorbent surface (1 - θ), and the desorption rate is proportional to the adsorbed amount (θ) dθ ¼ va - vd ¼ ka cð1 - θÞ - kd θ dt
ð1Þ
where θ = a/am is the relative surface coverage, a is the adsorbed amount, am is the adsorption capacity, t is time, and c is the temporary concentration, whereas ka and kd are the adsorption and desorption rate coefficients, respectively. When we take into account the Langmuir isotherm equilibrium coverage θeq,1,6,7 Kceq θeq ¼ ð2Þ 1 þ Kceq we obtain the rate equation dθ ¼ ka ðco - βθÞð1 - θÞ - kd θ dt
ð3Þ
where β ¼
mam co - ceq ka ¼ and K ¼ V θeq kd
ð4Þ
In the above equations, co and ceq are the initial and equilibrium concentrations, respectively, K is the Langmuir isotherm equilibrium constant, m is the adsorbent mass, and V is the solution volume. The integrated solution of the Langmuir rate equation obtained by Azizian was analytical; however, its complicated form does not allow for easy data presentation and analysis. He treated the rate equation as a common binomial of temporary surface coverage θ and solved it by using some substitutions, obtaining an equation equivalent to the one below: " ! !# 1 2βθ 2βθ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln 1þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi - ln 1þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ka t B2 - 4βco B - B2 - 4βco B þ B2 - 4βco
ð5aÞ 1 B ¼ - ðβ þ co þ Þ and B < 0 K
ð5bÞ
(63) Marczewski, A. W. Proceedings of the 10th Ukrainian-Polish Symposium on Theoretical and Experimental Studies of Interfacial Phenomena and Their Technological Applications; Lviv-Uzlissia, Ukraine, Sept 26-30, 2006; part 1, pp 237-239 (ISBN 966-613-442-X).
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Moreover, by using approximations, Azizian showed that the first-order model was obtained when the final uptake was very low (concentration did not change much during the course of experiment). However, the second-order model was obtained when the final uptake was high, but only the initial part of the kinetic experiment (low t and θ values) was considered in the derivation of the second-order kinetics. Later, Azizian extended his considerations by including the energetic heterogeneity corresponding to the Langmuir-Freundlich isotherm.64 From Azizian’s approximations we obtain apparent kinetic constants for the derived boundary solutions. (For definitions and relations of rate coefficients kia and kic, see Appendix A.) k1, app ¼ k1 ¼ k1c ¼ k1a ¼ ka co þ kd and k2, app ¼ k2 ¼ k2a ¼ ka
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 - 4βco - B 2θeq am
ð6Þ
ð7Þ
Mixed 1,2-Order Equation (MOE) Recently, Marczewski presented the application of a simple mixed 1,2-order equation (MOE) to the kinetics of methylene blue on mesoporous carbons.38 However, no specific physical model was considered in this paper despite mentioning its relation to the Langmuir kinetics,6,7 which may be considered to be binomial with respect to the adsorbed amount.8,9 The MOE was inspired by the observation that many systems show behavior intermediate between PFOE and PSOE. It may be written as dF ¼ k1a ð1 - FÞ þ k2a aeq ð1 - FÞ2 dt
ð8Þ
where kia represents the rate coefficients and F is the dimensionless adsorption progress (fractional attainment of equilibrium): F ¼
a aeq
ð9Þ
Equation 8 is a combination of the first-order and second-order rate terms38 and is similar in its form to the “hybrid-order” equation derived from the Langmuir kinetics6,7 by Liu and Shen;8 however, their equation was made dependent on the adsorption coverage rather than the adsorption progress and thus the resulting equations are much more difficult to analyze. After simple substitutions in eq 8, we obtain dF ¼ ðk1a þ k2a aeq Þ½ f1 ð1 - FÞ þ f2 ð1 - FÞ2 dt
If only the second-order contribution coefficient f2 0 fi
ð11Þ
(64) Azizian, S.; Haerifar, M.; Basiri-Parsa, J. Chemosphere 2007, 68, 2040– 2046.
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ð13aÞ
ð13bÞ
If the contribution of the second-order kinetic term in the rate expression becomes zero ( f2 = 0), then the Lagergren (i.e., the pure first-order equation (FOE or PFOE)) is obtained ( f1 = 1): lnð1 - FÞ ¼ -k1 t
ð14aÞ
a ¼ aeq ½1 - expð-k1 tÞ
ð14bÞ
If the contribution of first-order kinetics is zero ( f1 = 0) (i.e. f2 = 1), then the mixed 1,2-order rate equation (MOE, eq 8) becomes a pure second-order equation38 and the corresponding integrated form is the standard (pseudo)secondorder equation (PSOE):31 a ¼ aeq
k2a aeq t 1 þ k2a aeq t
ð15Þ
This equation is usually used in one of its various linear forms.23,25 The standard form commonly used for data fitting, the estimation of equilibrium adsorption, and the validation of PSOE applicability is26 t 1 t ¼ þ a k2a aeq 2 aeq
ð16Þ
Recently, Marczewski et al. showed38,56,57 that, even if used with caution, this linear form is overly optimistic in suggesting the overall applicability of PSOE;3,38,56,65 however, many authors use this linear form even if their data contains only values that are very close to equilibrium and such fitting may prove nothing. As was suggested in the literature,38,56,57 one of the alternative linear forms of the PSO equation23,25 is much better as an applicability control tool: a ¼ aeq -
¼ ðk1a þ k2a aeq Þ½ð1 - FÞð1 - f2 FÞ
ð12Þ
1 a k2a aeq t
ð17Þ
Plotting the kinetic data in a versus (a/t) coordinates exposes all discrepancies from the pure second-order (and to some extent first-order) equation and (with rare exceptions) makes it possible to show the entire kinetic adsorption curve from 0 to aeq in a compact graphic form with physically justified maximum values for both axes.38,56,57 Another advantage of this plot is that one of the axes is linear with respect to the experimental adsorption data (and also concentration because a ≈ (co - c)). In the case of the standard plot (eq 16), the x axis is time; however, values on both axes are not limited by any physical constraints, and with the increase in measurement time, only the asymptotic behavior close to the equilibrium range becomes important.3,38,56,57 (65) Rudzinski, W.; Pzazinski, W. Adsorption 2009, 15, 181–192.
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The integrated form of the MOE equation, having a form somewhat similar to the Lagergren (i.e., the PFO equation may describe a much wider spectrum of kinetic behaviors then the original FOE and SOE; also the systems have properties intermediate between those of the pure first- and second-order kinetics38). By using the alternative second-order plot (eq 17) instead of the standard plot (eq 16), it was shown that the MOE equation may describe rate behaviors that are impossible for the often preferred second-order equation (eq 15).38 However, its simple form made it possible to believe that the Langmuir rate equation (eq 1)1,6,7 may also have a similarly simple representation.
Integrated Kinetic Langmuir Equation: Analytic Solution of the Langmuir Rate Equation Azizian’s solution of the Langmuir rate equation9 was very important; however, it failed to suggest an easy-to-use form and thus did not change the typical approach that involved using PFOE and PSOE alternatively or some of their empirical modifications such as the MPFO equation.39 However, startlingly small changes in the approach led to a much simpler solution. The Langmuir rate equation includes changes to two domains: adsorption and concentration. Relative changes in adsorption are given by the adsorption progress F (eq 9); moreover, let us introduce a relative change in concentration (i.e. the uptake u and equilibrium uptake ueq): u ¼ 1-
c ceq and ueq ¼ 1 co co
ð18aÞ
u ¼ ueq F
ð18bÞ
At equilibrium, F = 1 and u = ueq. Hence c - ceq ¼ co ueq ð1 - FÞ and aeq ¼
Vco ueq m
ð19Þ
Let us rearrange the basic Langmuir rate equation (eq 1) by using the relative values of the equilibrium adsorption and initial concentration (i.e.,the adsorption progress F (eq 9) and equilibrium uptake ueq (eq 18): dθ ¼ ka co ð1 - ueq FÞð1 - Fθeq Þ - kd Fθeq dt θeq ¼ ka co ð1 - ueq FÞð1 - Fθeq Þ F Kco
Kceq Kco ð1 - ueq Þ ¼ 1 þ Kco ð1 - ueq Þ 1 þ Kceq
ð20Þ
ð21Þ
then θeq θeq Kco ð1 - ueq Þ ¼ and ð1 - θeq Þð1 - ueq Þ ¼ 1 - θeq Kco
ð22Þ
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ð23Þ
ð24Þ
Finally, by comparing with the MOE rate equation (eq 12), we can see that its form is identical to that of the differently defined rate coefficient and we may obtain this rate equation expressed by using the first-order kinetic coefficient k1 dF k1 ¼ ð1 - FÞð1 - ueq θeq FÞ dt 1 - ueq θeq ¼ k1 ð1 - FÞ
1 - ueq θeq F 1 - ueq θeq
ð25Þ
where ueqθeq
