Research: Science and Education
Derivation of the Freundlich Adsorption Isotherm from Kinetics Joseph Skopp School of Natural Resources, University of Nebraska, Lincoln, NE 68583;
[email protected] Adsorption of solutes on solid surfaces is a basic chemical process. A quantitative description of this process is essential to model the solute transport. The relation between the amount of solute in solution and the amount adsorbed is needed. The linear adsorption isotherm is the simplest model, S = KD C
(1)
where S is the amount adsorbed, C the concentration in solution, and KD the distribution coefficient. Equation 1 is readily derived from kinetic principles and the value of KD is understood to be the ratio of rate constants, KD =
kf kr
(2)
where kf is the forward rate constant from a first-order reaction and kr is the reverse rate constant. The interpretation of KD including its expected temperature dependence is straightforward. Unfortunately, many adsorption phenomena are not described by eq 1. Alternatives include the Freundlich and Langmuir isotherms,
S = KC b
S =
Km C k + C
Freundlich
(3)
Langmuir
(4)
where K, b, k and Km are fitting constants. The Langmuir isotherm is readily derived from kinetic principles (1) and its constants interpreted in terms of either rate constants or a maximum adsorption capacity. Historically, the Freundlich isotherm has been presented as an empirical equation with no, or limited, ability to interpret the coefficients. It is still common in environmental chemistry textbooks to represent the Freundlich isotherm in its historical context as an empirical equation. A sampling of some texts is instructive: ...the Freundlich equation is not theoretically based... (1, p 343) The equation has no theoretical foundation... (2, p 270) The Freundlich equation, ... is an empirical adsorption model (3, p 151)
However, a variety of alternative derivations of the Freundlich equation have been proposed. One of the earliest derivations assumed a distribution of adsorption sites (4) and this has been noted by some authors (5, 6). Unfortunately the derivation is difficult to apply or interpret and infrequently used. Most of these early efforts focus on thermodynamics of adsorption and emphasize energetics of surface heterogeneity. This allows the parameters of the Freundlich isotherm to be interpreted as a measure of the variation of site energies. Other derivations of the Freundlich isotherm have used a statistical mechanical approach
(9, 10). This approach has resulted in predictive models of the Freundlich isotherm parameters (9). One particularly simple prediction is that the parameter b from eq 3 should be inversely dependent on absolute temperature. Another equation that has received some attention is a modified Langmuir adsorption isotherm (sometimes called a fractal Langmuir). This equation was derived by Sposito (6) and used by Goldberg and Sposito (7) and Kano et al. (8):
S =
β KC
b
1 + KC Cb
(5)
Here b and K are constants distinct from the fitting constants used in eq 3 and β is a fitting constant. More recent approaches focus on the geometry of the adsorbent surface or of the porous material. These fractal approaches assume that adsorption is a time-dependent phenomenon with a slow diffusion step (either on the irregular surface or within micropores). This has also provided an explanation for hysteretic adsorption isotherms (11). Distinct derivations result in the same or similar equations. This creates ambiguity in the interpretation of data fit to these equations. Thus it is difficult to assign reaction mechanisms based on a single adsorption isotherm. This article describes a relatively simple derivation of the Freundlich isotherm and the fractal Langmuir isotherm using a kinetic approach. This derivation focuses on adsorption for porous materials. The use of fractals provides a diffusion-based picture of adsorption. The approach also provides a physical interpretation of the parameters in terms of molecular characteristics of the surface and adsorbate. More important, the approach is accessible to students with a limited background in kinetics. Results The starting point is the introduction of fractal kinetics. This can be introduced as a variation on integer-order kinetics. Typically first-order kinetics is described in introductory chemistry classes as follows:
dC = kC dt
(6)
where C is the concentration as a function of time, t, and k is the first-order rate constant. In general, the rate law can be written as
dC = kfr C n dt
(7)
where n is the order of the reaction, which may be a noninteger, and kfr is the fractal rate coefficient (12, 13). Such a rate law has been recognized for specific fractional reaction orders (e.g., 1/2, 3/2) both in traditional chemical kinetics as well as in en-
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 11 November 2009 • Journal of Chemical Education
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Research: Science and Education
vironmental chemistry (14). Here we are generalizing the order to any real number. Equation 7 is the basis for the derivation of the Freundlich isotherm. Understanding the basis of eq 7 is valuable. If it is treated as an empirical rate law, then the empirical nature of the Freundlich isotherm has only been hidden in a previous step in its derivation (albeit a still more satisfying approach than starting from eq 3). Instead we proceed by recognizing that adsorption in porous media (or on heterogeneous surfaces) is distinct from adsorption on planar surfaces. The distinction is that in porous media not all adsorption sites have equal access to the bulk solution. If we imagine a small pore (open at both ends), adsorption sites near the open ends are distinct from those at the center of the pore. This geometrical distinction occurs even if the sites possess uniform chemical properties. If a concentration is imposed at one end of the pore, solute must diffuse to find an open adsorption site as the sites near the open ends are occupied first. The desorption case is similar (i.e., solute must diffuse out from the center of the pore). The use of a fractal rate law assumes “diffusion-controlled reactions with geometrical constraints...may be described by reactions on fractal domains” (13). In this view, the exponent n is a measure of the fractal dimension of the process. Classical first-order rate constants have long been known to depend on the molecular diffusion coefficient (15). The rate constant for fractal reactions is also expected to be proportional to the effective molecular diffusion coefficient (16). Our derivation of the Freundlich isotherm proceeds by writing eq 6 for both the forward (adsorption) and backward (desorption) reactions but with distinct fractal dimensions (ni) and fractal rate constants (ki):
dC = −k1C n1 + k2 S n2 dt
(8)
At equilibrium the derivative is equal to zero. The equation can then be solved for S:
S =
k1 k2
1 n2
C n1
n2
(9)
This is identical to the standard Freundlich isotherm, eq 3, with K = (k1/k2)1/n1 and b = n1/n2. Equation 9 provides a theoretical basis for interpreting the constants obtained from the Freundlich isotherm. A similar derivation is possible for the modified Langmuir adsorption isotherm, eq 5, presented by Goldberg and Sposito (7). It requires the assumption of a limiting adsorption capacity and that the desorption reaction is linear rather than fractal,
dC = −k1 C n1 ( S0 − S ) + k2 S dt
(10)
where S0 is the maximum adsorption capacity. Setting the derivative equal to zero and solving for S yields:
S =
(k1
k2 ) S0 C n1
1 + (k1 k2 ) C n1
(11)
This is identical to the standard modified Langmuir adsorption isotherm, eq 5, with K = (k1/k2), β = S0, and b = n1. This model 1342
can be extended to include fractal desorption but the mathematical solution must be obtained either as an approximation or numerically. Discussion The derivation of the Freundlich isotherm and fractal Langmuir isotherm provides new insights about the adsorption process. This includes both predictive as well as explanatory power. From a predictive point of view it first suggests how adsorption isotherm constants will change with diffusion coefficients, D, (k is proportional to D) or temperature (Arrhenius relation, k = A e –E/RT where A is the frequency factor, E the activation energy, R the gas constant, and T the absolute temperature). A second prediction is the possibility that reaction orders and rate constants may depend on sample size. This suggests the possibility of scale-dependent adsorption properties (because diffusion is scale dependent). While a great deal of work has gone into examining the scale dependence and fractal nature of the physical properties of porous materials (17–19), less effort has gone into such behavior for chemical properties (20). This has implications for the design of an experiment such as presented by Grubbs in this Journal (21). Here, different adsorbent masses are used that may represent an uncontrolled variable. A third prediction is the time dependence of linear adsorption isotherm constants as a reflection of time dependence of the rate constants. Integration of eq 7 (where C n−1 is lumped with the rate constant) results in an equation that if treated “as if it were first-order kinetics” possesses a time-dependent rate constant. This would influence the multiplicative constant rather than the exponent in eq 9. Kuo and Lotse (22) have observed this time-dependent behavior. The explanatory power of the model requires some insight into the nature of the fractal order of the reaction (ni from eq 9). These exponents can be related to the spectral dimension (ds) of the media. The spectral dimension is a property that describes the accessibility of surfaces (12) and the probability distribution for a molecule to access adsorption sites. The specific relation of the fractal reaction order and the spectral dimension is an area of study (e.g., see ref 23). In general as the ratio n1 over n2 approaches one, the spectral dimension can be expected to approach 2 (for reaction on an ideal planar surface). Conversely, as the ratio of n1 to n2 approaches zero, the spectral dimension approaches 1. Thus, the exponent for the Freundlich isotherm, n1/n2, is a measure of the deviation from an infinite perfectly plane homogeneous surface in contact with a well-mixed solution. Reductions in the exponent from 1 indicate greater spatial constraints on adsorption. This is a reflection of particle surface 3D morphology and the ability of the adsorbing molecule to access this surface. The key test of a physically constrained adsorption process is to look for the time dependence of the adsorption isotherm. In many environmental chemistry applications it is well known that adsorption from aqueous solutions may change slowly with time. Consequently the determination of many isotherms is constrained by the practicality of relatively short equilibration (typically 24–48 hours). The observance of a Freundlich isotherm or of hysteresis between adsorption and desorption isotherms may indicate that the isotherm represents quasi-equilibrium. A good example of these changes is shown in a recent article by Ahmed et al. (24). They showed changes in cadmium
Journal of Chemical Education • Vol. 86 No. 11 November 2009 • www.JCE.DivCHED.org • © Division of Chemical Education
Research: Science and Education
sorption and desorption over the course of 210 days. They also compared a purely diffusion model to a combined kinetic–rapid fractal Freundlich equilibrium. The use of time-dependent data provided the means to distinguish between these models and infer possible mechanisms. Hysteresis between adsorption and desorption isotherms may result from multiple mechanisms. Examples of mechanisms resulting in such behavior include co-precipitation, solid-state diffusion, in addition to constrained-surface diffusion. The reversibility of these mechanisms may not be identical so that more detailed kinetic models of the desorption step may provide a means of distinguishing among them. Extension of these results to multiple sorption sites or limiting capacity (e.g., Langmuir) is not as straightforward. In the first case the result is a nonlinear algebra equation that must be solved by root finding numerical algorithms. In the second case the use of fractal kinetics hides the detailed physical information needed to define a maximum capacity (e.g., surface area of a fractal surface is not well defined). Conclusions A simple derivation of the Freundlich adsorption isotherm is presented. This derivation can easily be presented in introductory or applied chemistry classes. The derivation allows the constants to be interpreted in terms of more fundamental kinetic or diffusion based properties. Predictions about the Freundlich isotherm constants can also be made. Thus, while the Freundlich isotherm is commonly used in an empirical manner, it should no longer be referred to as “merely an empirical equation”. Literature Cited 1. Essington, M. E. Soil and Water Chemistry: An Integrative Approach; CRC Press: Boca Raton, FL, 2004. 2. Tan, K. H. Principles of Soil Chemistry, 3rd ed.; Marcel Dekker, Inc.: New York, 1998. 3. Sparks, D. L. Environmental Soil Chemistry, 2nd ed.; Academic Press: San Diego, 2003. 4. Sips, R. J. Chem. Phys. 1948, 16, 490–495. 5. Thomas, J. M.; Thomas, W. J. Introduction to the Principles of Heterogeneous Catalysis; Academic Press: London, 1967. 6. Sposito, G. Soil Sci. Soc. Am. J. 1980, 44, 652–654.
7. Goldberg, S.; Sposito, G. Soil Sci. Soc. Am. J. 1984, 48, 772– 778. 8. Kano, F.; Abe, I.; Kamaya, H.; Ueda, I. Surface Science 2000, 467, 131–138. 9. Yang, C. J. Colloid Interface Sci. 1998, 208, 379–387. 10. Liu, Y. H. X.; Joo-Hwa, T. J. Environ. Eng. 2005 131, 1466– 1468. 11. Seri-Levy, A.; Avnir, D. Langmuir 1993, 9, 3067–3076. 12. Kopelman, R. Science 1988, 241, 1620–1626. 13. Kopelman, R. Diffusion-Controlled Reaction Kinetics. In The Fractal Approach to Heterogeneous Chemistry; Avnir, D., Ed.; John Wiley and Sons: Chichester, U.K., 1989; Chapter 4.1.3. 14. Sparks, D. L. Kinetics of Soil Chemical Processes; Academic Press: San Diego, 1989. 15. Smoluchowski, M. V. Z. Phys. Chem. 1917, 92, 129–168. 16. Kopelman, R. J. Stat. Phys. 1986, 42, 185–200. 17. Fractals in Soil Science; Baveye, P., Parlange, J., Stewart, B. A., Eds.; CRC Press: Boca Raton, 1998. 18. Pachepsky, Y. A.; Crawford, J. W.; Rawls W. J. Fractals in Soil Science; Developments in Soil Science 27; Elsevier: Amsterdam, 2000. 19. Pachepsky, Y.; Radcliffe, D. E.; Selim, H. M. Scaling Methods in Soil Physics; CRC Press: Boca Raton, 2003. 20. Okuda, I.; Senesi, N. Fractal Principles and Methods Applied to the Chemistry of Sorption onto Environmental Particles. In Structure and Surface Reactions of Soil Particles; Huang, P. M., Senesin N., Buffle, J., Eds.; John Wiley and Sons: Chichester, U.K., 1998. 21. Grubbs, T. JCE Data-Driven Exercises. http://www.jce.divched. org/JCEDLib/DataDriven/index.html and http://www.stetson. edu/~wgrubbs/datadriven/langmuir/langmuirwtg.html (both accessed Jul 2009). 22. Kuo, S.; Lotse, E. G. Soil Sci. 1973, 116, 400–406. 23. Redner, S.; Kang, K. Phys. Rev. A 1984, 30, 3362–3365. 24. Ahmed, I. A. M.; Crout, N. M. J.; Young, S. D. Geochim. Cosmochim. Acta 2008, 72, 1498–1512.
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