pubs.acs.org/Langmuir © 2009 American Chemical Society
A Novel Two-Resistance Model for Description of the Adsorption Kinetics onto Porous Particles Wojciech Plazinski*,†,‡ and Wzadyszaw Rudzinski†,‡ †
Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Cracow, Poland, and ‡Department of Theoretical Chemistry, Faculty of Chemistry, UMCS, pl. M. Curie-Sklodowskiej 3, 20-031 Lublin, Poland Received June 19, 2009. Revised Manuscript Received July 22, 2009
A novel two-resistance model for description of the sorption kinetics by porous particles has been proposed. The model takes into account two kinetic steps of different kinds which are involved in the overall sorption process rate: (i) the rate of solute diffusion in pores of the sorbent particles having uniform sizes and characterized by the homogeneous intraparticle diffusion coefficient and (ii) the rate of a direct adsorption/desorption process on the surface, described by applying the statistical rate theory (SRT) approach. Two different kinds of sorbent particles geometry are considered: the spherical and the plane particles, having their dimension characterized by radius and thickness, respectively. The meaning of the parameters which influence the sorption kinetics has been discussed. The results make it possible to judge which conditions have to be fulfilled to consider only one kinetic step and, thus, to simplify the theoretical description of a given system. The conclusion has been drawn that the concave character of kinetic sorption isotherms plotted in the function of the square root of time is of a general nature and is connected with the situation when at least two different processes are involved in controlling the sorption kinetics.
1. Introduction A sorption process at the solid/solution interface can be described by the following four consecutive stages :1 1 transport of sorbate in the bulk solution; 2 diffusion across the film surrounding the sorbent particles (diffusion in the subsurface region); 3 migration of sorbate within the pores of the sorbent (intraparticle diffusion); 4 adsorption/desorption on the solid surface viewed as a kind of chemical reaction (surface reaction). It is most often assumed that only one of these steps controls the overall rate of sorption, as the slowest one.1 This assumption seems to be the crucial one when considering the theoretical description of the sorption kinetics. Different choices of the ratelimiting step have resulted in construction of many theoretical models. Among them, the classical Langmuir kinetic model of adsorption,2,3 statistical rate theory approach,4-7 intraparticle homogeneous diffusion model,8,9 shrinking core model,10 homogeneous surface diffusion model,11 and diffusion in the subsurface region model 12 can be mentioned. These general models might be used as a basis for developing more sophisticated and complicated models, taking into account more than only *Corresponding author. Fax: +48-81-537-5685; e-mail: wojtek_plazinski@ o2.pl. (1) Ho, Y. S.; Ng, J. C. Y.; McKay, G. Sep. Purif. Methods 2000, 29, 189. (2) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (3) Azizian, S. J. Colloid Interface Sci. 2004, 276, 47. (4) Rudzinski, W.; Plazinski, W. J. Phys. Chem. B 2006, 110, 16514. (5) Rudzinski, W.; Plazinski, W. J. Phys. Chem. C 2007, 111, 15100. (6) Azizian, S.; Bashiri, H. Langmuir 2008, 24, 13013. (7) Azizian, S.; Bashiri, H. Langmuir 2008, 24, 11669. (8) Suzuki, M. Adsorption Engineering; Kodansha: Tokyo, 1990. (9) Evans, J. R.; Davids, W. G.; MacRae, J. D.; Amirbahman, A. Water Res. 2002, 36, 3219. (10) Juang, R.-S.; Lin, H.-C. J. Chem. Technol. Biotechnol. 1995, 62, 132. (11) Chen, J. P.; Yang, L. Ind. Eng. Chem. Res. 2005, 44, 9931. (12) McKay, G.; Allen, S. J. Can. J. Chem. Eng. 1980, 58, 521.
802 DOI: 10.1021/la902211c
one rate-limiting step,1,5,13 for instance, or many different factors, such as pH14 or the presence of competitive solutes.15 Application of diffusional models is usually connected with accepting the assumption of a local equilibrium between the solute in the solution and those adsorbed on the surface. This is justified only when the rate of surface reaction is much faster than that of diffusion-driven transport of solute. The models in which the rate of surface reaction is also taken into account were developed primarily to predict the rate of solute migration in the groundwater systems.16-18 Similar considerations are not common in the case of the systems designed to remove pollutants from aqueous solution. Moreover, the existing models are usually based on the simplest possible kinetic equations (first order reversible reaction occurring on the sorbent surface, for instance). Such approach is not able to reflect the effects of surface saturation and the nonlinear character of the equilibrium adsorption isotherm. These facts justify the reasons for developing new models, based on some more general assumptions. In our previous papers we have proposed two methods of combining the rate of surface reaction with that of diffusion in the subsurface region19 or with the rate of intraparticle diffusion.5,20 In both cases the rate of surface reaction was expressed in terms of the new approach to interfacial kinetics, called the statistical rate theory (SRT). Models based on accepting the intraparticle diffusion as the most essential process and, additionally, incorporating the rate of surface reaction are rather rarely found in literature. Our previous attempts were directed toward approximating the initial and the final regions of the kinetic (13) Choy, K. K. H.; Ko, D. C. K.; Cheung, C. W.; Porter, J. F.; McKay, G. J. Colloid Interface Sci. 2004, 271, 284. (14) Rudzinski, W.; Plazinski, W. J. Colloid Interface Sci. 2008, 327, 36. (15) Azizian, S.; Bashiri, H.; Iloukhani, H. J. Phys. Chem. C 2008, 112, 10251. (16) Bahr, J. M.; Rubin, J. Water Resour. Res. 1987, 23, 438. (17) Barry, D. A.; Prommer, H.; Miller, C. T.; Engesgaard, P.; Brun, A.; Zheng, C. Adv. Water Resour. 2002, 25, 945. (18) MacQuarrie, K. T. B.; Mayer, K. U. Earth-Sci. Rev. 2005, 72, 189. (19) Rudzinski, W.; Plazinski, W. Langmuir 2008, 24, 6738. (20) Rudzinski, W.; Plazinski, W. Environ. Sci. Technol. 2008, 42, 2470.
Published on Web 08/13/2009
Langmuir 2010, 26(2), 802–808
Plazinski and Rudzinski
Article Table 1. Basic Mathematical Expression Associated with the IDM Approach
basic equation
boundary conditions
initial condition averaged amount adsorbed
analytical solution
spherical particles ! Dc D D2 c 2 Dc F Dq ¼ þ Dt ι Dr2 r Dr εP Dt Dc ¼0 Dr r ¼0 and c(r = r0) = cb cð0erer0 Þ ¼ 0 at t ¼ 0 qðtÞ ¼
3 r30
Z
r0
r2 qðr, tÞdr
ð1Þ
Dc D D2 c F Dq ¼ Dt ι Dx2 εP Dt
ð7Þ
ð2Þ
Dc ¼0 Dx x ¼0
ð8Þ
ð3Þ
and c(x = ( l) = cb cð -lexelÞ ¼ 0at t ¼ 0
ð4Þ
qðtÞ ¼
0
! ¥ qðtÞ 6 X 1 Da π2 n2 ¼ 1- 2 exp t ð5Þ π n ¼1 n2 qe r20
isotherm by equations arising from the surface reaction model and intraparticle diffusion model, respectively. In the present work, we are going to develop a more general method, which takes into account both (i) the rate of solute diffusion in pores of the sorbent particles having uniform sizes and homogeneous intraparticle diffusion coefficient value and (ii) the rate of direct adsorption/desorption process on the surface, expressed by using the SRT approach. Two different kinds of sorbent particles geometry were considered, namely: the spherical and the plane particles, having their dimensions characterized by certain values of radius and thickness, respectively. The second objective of this paper is to discuss the parameters that influence the sorption kinetics and are connected with both the surface reaction and intraparticle diffusion models. Detailed knowledge of this issue is required to state which of the conditions have to be fulfilled to consider only one kinetic step and simplify the theoretical description of a given system.
2. The Principles of the Intraparticle Diffusion Model (IDM) and the Statistical Rate Theory (SRT) The description of sorbate intraparticle diffusion and adsorption onto the porous particle surface is based on the Fick’s laws of diffusion. Its principles are very well-known in literature 8,13,21 but for further consideration and comparison, let us recall the most fundamental issues. The most important mathematical expressions related to this approach are collected in Table 1. The sorbent particles are most often modeled as spheres with certain properties such as: particle porosity (εP), particle density (F) and the so-called tortuosity factor (ι). Then the time-evolution of the process can be generally described by the mass balance eq 1 using the radial coordinate r in which t is the time and D is the molecular diffusion coefficient of sorbate in the liquid solution. The local concentrations of the sorbate within a particle in the solution, c, and in the adsorbed phase, q are related to their equilibrium relationship; cb is the solute concentration in the bulk solution. The boundary conditions are given by eqs 2 and 3. The initial condition for eq 1 corresponding to the experimental conditions (adsorption process) is expressed by eq 3. The average amount adsorbed onto the particle surface, q, can be found by applying the appropriate equilibrium adsorption isotherm equation and then, by averaging the amount adsorbed in a given point of the particle over the volume of the particle,8,21 as (21) Crank, J. Mathematics of Diffusion., 2nd ed; Clarendon Press: Oxford, UK, 1975.
Langmuir 2010, 26(2), 802–808
plane particles
1 l
Z
ð9Þ
l
ð10Þ
qðx, tÞdx 0
¥ qðtÞ 8 X 1 Da π2 ð2n þ 1Þ2 ¼ 1- 2 exp t π n ¼0 ð2n þ 1Þ2 qe 4l 2
! ð11Þ
shown by eq 4. Of course, the equilibrium adsorbed amount is qe = q(t f ¥). The model described above and its analogues, related to particles having different geometry, are together referred to as the “intraparticle diffusion model” (IDM) here. Under some idealized assumptions (i.e., all particles are of identical dimension, the sorbate concentration in the solution is constant and the Henry’s equation is applicable for the considered system), eq 1 can be solved analytically.8,21 Then, the kinetic adsorption isotherm can be expressed by 5, in which the Da constant is called the effective (apparent) intraparticle diffusion coefficient:8 Da ¼
D ιð1 þ FKH =εP Þ
ð6Þ
where KH is the Henry constant. One can also consider the sorbent particles having a plane shape. This kind of geometry might be particularly interesting when taking into account some biosorption systems.22,23 The appropriate equation has the form given by eq 7, in which x is the arbitrary position coordinate from the central line of the particle in the thickwise direction. The boundary 8 and initial 9 conditions for the adsorption process are analogical to those introduced for spherical particles. The problem considered here is symmetrical about the central plane of the particle; thus, it is the most convenient to set the surfaces at x = ( l. Thus, the thickness of the particle is equal to 2l. The average adsorbed amount can be calculated from relation 10. Under conditions similar to those mentioned above for the case of spherical particles eq 7 can be solved analytically to yield 11, where the apparent diffusion coefficient Da is expressed by relation 6 once more. Both eqs 5 and 11 were developed under the assumption that the rate of diffusion is much slower than that of direct adsorbate transfer from the bulk to the adsorbed phase (the latter will be called “the rate of surface reaction”). As the local equilibrium was assumed to exist, the accepted equilibrium isotherm equation could be used for calculating the q(x,t) or q(r,t) values. In the next section the issue of taking into account both the rate of intraparticle diffusion and the rate of surface reaction will be investigated. The SRT approach will be applied for this purpose. The Statistical Rate Theory of Interfacial Transport was first proposed in the literature by Ward et al.24 The SRT approach links (22) Yang, J.; Volesky, B. Environ. Sci. Technol. 1999, 33, 751. (23) Davis, T. A.; Volesky, B.; Mucci, A. Water Res. 2003, 37, 4311. (24) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599.
DOI: 10.1021/la902211c
803
Article
Plazinski and Rudzinski
the rate of transport between two phases with different chemical potentials of the considered species in these phases. It has been successfully applied to describe the rates of interfacial transports of various kinds. Recently it has been also used to describe the kinetics of adsorption at the solid/solution interfaces.4-7,14,15,19,20 In our previous papers we have developed the SRT equations describing the rate of adsorption on the energetically homogeneous solid surface, represented by the classical Langmuir model.4,5 We have also shown how to incorporate the effect resulting from the surface energetic heterogeneity into the SRT model.4 Our considerations will be limited here to the case of an adsorption system which can be described by the model of homogeneous surface and the Henry and Langmuir isotherm equations. Based on our previous results, it can be easily proved that, under such assumption, the SRT rate equations take the following explicit forms: dq KH c b q ð12Þ ¼ Kls ce dt KH c b q dq ðqm -qÞ 1 q ¼ Kls ce ð1 -qe =qm Þ KL cb dt KL cb ðqm -qÞ q
ð13Þ
where eqs 12 and 13 correspond to the Henry and the Langmuir equations, respectively; ce is the equilibrium solute concentration in the bulk phase, qm is the monolayer capacity and Kls is a constant. KL is the parameter of the Langmuir equation, i.e., the Langmuir constant. Moreover, let us introduce the following rate equation, which has previously been combined with the IDM16 (and called the first-order reversible reaction model 1): dq ¼ K1 cb -K2 q dt
ð14Þ
in which K1 and K2 are the constants corresponding to the adsorption and desorption rates, respectively. The above expression can be interpreted as a special case of the Langmuir kinetic model, in which the effect of surface saturation is not considered. Thus, eq 14 is valid only for diluted solution, and, similarly to eq 12, corresponds to the Henry adsorption isotherm equation in which KH = K1/K2. This formula is introduced for its comparison with the SRT-related expressions 12 and 13. An additional assumption is that the solute concentration in the bulk phase is constant throughout the kinetic study. In other words, the relation cb = ce = cin = const is fulfilled, in which cin is the initial sorbate concentration in the solution. It should be emphasized that eqs 12 and 13 were derived under the assumption that the rate of intraparticle diffusion is much faster than that of surface reaction. All other kinds of kinetic mechanisms originating from the family of diffusional models (such as external-film-diffusion model, for instance) were also neglected. Thus, the SRT approach does not take into account the internal structure of the adsorbent, expressed by its porosity or by the dimensions of particles, which is crucial for all diffusional models. However, some of these models can be combined with SRT in an approximate way, as shown in our previous papers.5,20 Now we are going to focus on the exact method of connecting both the IDM and the SRT approaches.
3. Mathematical Formulation of the Two-Resistance Kinetic Model The model proposed here takes into account both the rate of intraparticle diffusion expressed by eq 1 (or eq 7; the choice 804 DOI: 10.1021/la902211c
depends on the preferred geometry of the particles) and the rate of surface reaction, expressed by using the SRT approach. The most intuitive method for solving this issue is based on transforming the average adsorbed amount q(t), appearing in eqs 12-14, into its “local” equivalent, i.e., q(r,t) (or q(x,t)) and writing the appropriate set of partial differential equations. The bulk solution concentration variable, cb, should be treated in a similar way, i.e. cb f c. Further, the value of the ce parameter, appearing in eqs 12 and 13 should not be, in general, equal to cin. When considering eqs 12 and 13, ce corresponds to the concentration to which the system evolves when reaching an equilibrium. In the case of the two-resistance model, it depends on both coordinates and t and can be approximated by c as it is the value to which the concentration in a given point inside the sorbent particle would evolve if the rate of diffusion were much faster than that of surface reaction. In order to simplify the further calculations, one can introduce the following dimensionless variables: 8 > > > R ¼ r=r0 > > > X ¼ x=l > > > > τ ¼ Dr20 t=ι or τ ¼ Dl 2 t=ι > > > > > C ¼ c=cin > > > > > Q ¼ q=qe > > > > Fq > Z ¼ qm =qe > > > > K2 ι K2 ι > > > φ0 ¼ 2 or φ0 ¼ 2 > > Dl Dr > 0 > > > Kls ι Kls ι > > φ1 ¼ or φ1 ¼ > > 2 2 > K Dr K H Dl H > 0 > > > K ι K ι > ls ls > or φ2 ¼ : φ2 ¼ qm Dl 2 qm Dr20
ð15Þ
Then, eqs 1 and 15 for the case of spherical particles, yield DCðR, τÞ D2 CðR, τÞ 2 DCðR, τÞ DQðR, τÞ þ ¼ -ξ 2 Dτ DR R DR Dτ
ð16Þ
While for the case of plane particles one obtains DCðX, τÞ D2 CðX, τÞ DQðX, τÞ ¼ -ξ Dτ DX 2 Dτ
ð17Þ
Then, eq 14 and SRT-related expressions 12 and 13 can be written as eqs 18-20, respectively: DQ ¼ φ0 ðC -QÞ Dτ
ð18Þ
! DQ C2 ¼ φ1 -Q Dτ Q
DQ C2 ðZ -QÞ2 ¼ φ2 -QðZ -1Þ Dτ Q Z -1
ð19Þ ! ð20Þ
The assumption of the constant sorbate concentration in the bulk phase has been maintained when deriving eqs 18-20. The mathematical form of eqs 19-20 is related to those developed previously by Rudzinski and Panczyk for the gas/ solid adsorption systems and characteristic of the case of the Langmuir 2010, 26(2), 802–808
Plazinski and Rudzinski
“volume-dominated” systems.25 The above assumption leading to the substitution ce f c in eqs 19 and 20 is somewhat approximated, but the exact determination of the “apparent” ce value to which the c value inside the sorbent particle would evolve (if taking into account the rate of diffusion) is impossible. Nevertheless, the c value is closer to the actual one than the cin value, which would be appropriate for nonporous sorbents (for more details see ref 8). Let us note that φi constants play a role of time-scaling coefficients, i.e., the higher is their value, the faster is the rate of surface reaction. The Z constant represents the character of the equilibrium adsorption isotherm; it can be proved that in the limit Z f ¥, eq 20 reduces to eq 19. (This situation correspond to the similarity of the Henry and the Langmuir equations for low solute concentration). When the Z values approach unity, the nonlinearity of the equilibrium adsorption isotherm becomes more significant. Thus, considering both eqs 19 and 20 is justified only for relatively low values of Z. Simple calculation (data not shown) makes it possible to state that this situation occurs for Z < 10. The ζ coefficient represents the sorbent capacity, i.e., the higher is the ζ value, the better is the sorbent ability to remove the solute from the solution. Equations 16-20 are the mathematical formulation of the two-resistance kinetic model presented here. The corresponding initial and boundary conditions are analogical to those introduced previously.
Article
Figure 1. The comparison of the behavior of the kinetic isotherms Q(τ1/2) developed for the two resistance model by the combination of eq 18 (first order reversible kinetics) and eq 16, corresponding to the spherical sorbent particles. The parameters used in these calculations are given in the Figure.
4. Model Calculations and Their Results The Comsol Multiphysics software application (Comsol Multiphysics, Version 3.3) was used to solve the coupled sets of differential equations using a finite element method. The values of only two parameters (ζ and φi) in the case of each set of equations had to be assumed for this purpose. The actual value of ζ depends on both the type of sorbent, its internal structure, and the equilibrium features of the system containing this sorbent. At a starting point, we have considered the model system characterized by ζ = 200. As one can easily check, this value is reliable for both the typical biosorbents 26 and other types of sorbents.8 Determining the φi value is not an easy task, as it is related to the kinetic features of the system which can not be directly measured. One of the aims of this study was to estimate how large the φi coefficient value has to be to state that the contribution of the rate of surface reaction can be neglected when modeling the overall rate of the sorption process. On the contrary, very low values of φi result in the possibility of neglecting the rate of the diffusiondriven transport and one faces the situation described by eqs 18-20. The intermediate cases are the matter of the present studies. Furthermore, one can notice that the direct combination of eqs 16 or 17 with the appropriate “surface reaction” rate equation leads to the product ζφi in the term expressing the rate of adsorption. Thus, all the calculations performed for ζ = 200 can be easily generalized for any other cases, corresponding to different values of ζ. The ζ = 200 assignment was made in order to separate the “equilibrium” and “operational” factors represented by the ζ value from those describing only the kinetic features of the system, i.e., φi coefficients. The selected results of calculations are presented in Figures 1-5. The Q vs τ1/2 scale was chosen due to similar mathematical features of both IDM and SRT models. In our previous papers 5,20 we have shown that both these approaches predict the linear dependence of the adsorbed amount on the square root of time for initial times of sorption process. (This fact has been commonly (25) Rudzinski, W.; Panczyk, T. J. Non-Equilib. Thermodyn. 2002, 27, 149. (26) Veglio, F.; Beolchini, F. Hydrometallurgy 1997, 44, 301.
Langmuir 2010, 26(2), 802–808
Figure 2. The behavior of the kinetic isotherms Q(τ1/2) developed for the two resistance model by the combination of eq 19 (SRT approach and linear equilibrium adsorption isotherm equation) and eq 16, (IDM approach and spherical sorbent particles). The parameters used in these calculations are given in the Figure.
Figure 3. The behavior of the kinetic isotherms Q(τ1/2) developed for the two resistance model by the combination of eq 20 (SRT approach and nonlinear equilibrium adsorption isotherm equation, represented by Z = 1.25) and eqs 16, (IDM approach and spherical sorbent particles). Other parameters used in these calculations are given in the Figure.
known for the IDM approach and was used as a tool to test if the rate of sorption is controlled by intraparticle diffusion). 8 Further, it was shown that the concave character of this square-root dependence on time may be due to a combined effect of two DOI: 10.1021/la902211c
805
Article
Figure 4. The comparison of the behavior of the kinetic isotherms Q(τ1/2) developed for the two resistance model by the combination of eq 18 (first order reversible kinetics) and eqs 17, corresponding to the plane sorbent particles. The parameters used in these calculations are given in the Figure.
Figure 5. The behavior of the kinetic isotherms Q(τ1/2) developed for the two resistance model by the combination of eq 19 (SRT approach and linear equilibrium adsorption isotherm equation) and eqs 17, (IDM approach and plane sorbent particles). The parameters used in these calculations are given in the Figure.
different kinetic mechanisms involved in the sorption process (i.e., the rate of surface reaction and that of the transport from the bulk to the surface).19 Thus, introducing the mentioned scale of time has its justification. The most important observations can be summarized as follows: 1 According to the assumptions, the “combined” approach reduces to the SRT (when the φi value is low) or to the IDM (when φi is high) approaches, which are its limiting cases. Details can be seen in Figures 1-5. 2 The (approximate) ranges of values of φi below and above which the situation can be described by eqs 18-20 and 16 and 17, respectively, are following: φ0 ∈ (0.005; 1), φ1 ∈ (0.001; 0.5), φ2 ∈ (0.0005; 0.1). Apparently, these ranges are not dependent on the particles geometry, although, in general, the geometry influences the sorption kinetics limited by the intraparticle diffusion.8 3 In all the intermediate cases (when the φi value lies within the above-mentioned ranges) one can clearly observe the concave character of the Q(τ1/2) kinetic isotherms (see Figures 1-5). When combining this observation with those made in our previous paper,19 it is very probable that this behavior is of a general nature and occurs when the following conditions are fulfilled: (i) at least two different processes are involved in controlling the sorption kinetics; (ii) the 806 DOI: 10.1021/la902211c
Plazinski and Rudzinski
rates of these processes are of comparable value. The “comparable value” term can be explained by giving the values of certain coefficient (as it was done in point 2.). For more details see the results of model calculations presented in ref 19, where the case of the SRT approach combined with the “boundary layer mass transfer” model was discussed. In spite of the qualitative similarities in the course of the Q(τ1/2) functions there exists main difference which allows for distinguishing between these two kinds of “combined” mechanisms; in the case of “boundary layer mass transfer”, the concave character of the Q(τ1/2) isotherm should be strongly dependent on the agitation rate applied in the system. (More details can be found in refs 1,8,9, whereas the example of the related experimental data are presented in ref 27). Such dependence is not characteristic of the model presented here. 4 Introducing the nonlinearity (connected with the equilibrium adsorption isotherm equation) into the SRT expressions does not influence significantly the course of kinetic isotherms of sorption. This observation is more obvious when plotting isotherms as Q(τ) vs τ/τ(Q = 0.99) where τ(Q = 0.99) is the time at which Q = 0.99 (data not shown). However, the impact of the “nonlinearity” effect is evident when considering the time of equilibration, which is the direct consequence of the mathematical form of eq 20. 5 The assumed shape of sorbent particles has also only very limited influence on the course of Q(τ1/2) functions as we have not noticed any significant differences between the kinetic isotherms generated for spherical and plane particles for the same sets of parameters. As the values of φi coefficients influence the sorption kinetics (to a certain degree), they obviously are important when considering the equilibration time characteristic of a given system. As mentioned, an important premise for stating that the two-resistance mechanism exists is associated with the existence of the concavity in the Q(τ1/2) kinetic isotherm. Nevertheless, the initial range of kinetic sorption isotherm may not be monitored accurately enough to draw any reliable conclusions. Thus, studying the equilibration times may allow for stating if, apart from intraparticle diffusion, other mechanisms are involved in the sorption kinetics. The related calculations were performed for the case of spherical particles and eqs 18-20 expressing three different models of the surface reaction rate. A few selected results are presented in Figure 6. The log-log scale was chosen for better clarity, as the values of equilibration time as well as those of φi can vary by a few orders of magnitude. The equilibration time was defined as the value of time τ at which Q = 0.99 (and denoted as τ (Q = 0.99)). The results obtained by using only IDM as well as only eqs 18-20 are also presented for comparative purposes. The obtained results are in agreement with those already presented, i.e., the ranges of φi values for which τ(Q = 0.99) differs significantly from those predicted by IDM and SRT models are approximately the same as those mentioned above in point 2. Apparently, the differences observed in panel (A) of Figure 6 do not influence significantly the course of Q(τ1/2) kinetic isotherms when Q < 0.95, thus, it was assumed that for φ0 > 1 both the IDM and the “two-resistance model” approaches exhibit nearly identical behavior. This means, that the equilibration time characteristic of a given system can confirm the existence of the two-resistance kinetic mechanism controlling the sorption kinetics. (27) Chu, H. C.; Chen, K. M. Process Biochem. 2002, 37, 1129.
Langmuir 2010, 26(2), 802–808
Plazinski and Rudzinski
Article
Figure 6. The simulated values of equilibration time (denoted here as τ(Q = 0.99)) for the system containing the spherical sorbent particles. Solid (______) and broken (- - - -) lines represent the equilibration times calculated from IDM (eq 16) and “surface reaction” (eqs 18-20) approaches, respectively. Other details: (A) eq 18 was accepted (first order reversible kinetics). A difference between τ(Q = 0.99) calculated from the IDM approach and those simulated for φ0 = 1 is still visible. In spite of this, it can be safely assumed that for φ0 > 1 both approaches exhibit nearly identical behavior for Q < 0.95. This behavior is probably the result of the mathematical form of eq 18, as it is not present in the cases (B) and (C), which correspond to the SRT approach and eqs 19 and 20, respectively.
Figure 7. The results of applying the approximated procedure of analysis of the experimental data, proposed in ref 20. The fitted data (O) were generated by using eqs 16 and 19 for φ1 = 0.005, i.e., the intermediate case for which both the rates of surface reaction and the intraparticle diffusion play a significant role. The solid lines (______) are the plots obtained by using the approximated approach. (A) ζ, φ1 and τ0 are treated as the best-fit parameters. Their (obtained) values are: φ1 = 0.00281 and ζ = 1906. (B) The “composite” plot was obtained by using the same values of ζ and φ1 as those used for generating the fitted data.
Let us consider the model system (containing spherical sorbent particles and described by the Henry equilibrium sorption isotherm equation) characterized by the following values of the most important coefficients: ζ = 200, D/ι = 1.5 10-5 cm2/s and R0 = 0.35 mm. The assumed values are in agreement with those reported in literature for the biosorption systems.28 Then, if the intraparticle diffusion is the only process the rate of which is involved in the sorption kinetics, the equilibration time is equal to t(Q = 0.99) = 83 s. A much higher value of t(Q = 0.99) would suggest that some other processes may also play a role. A more detailed study is possible only for the case of more exactly defined sorption systems.
(28) Papageorgiou, S. K.; Katsaros, F. K.; Kouvelos, E. P.; Nolan, J. W.; Le Deit, H.; Kanellopoulos, N. K. J. Hazard. Mater. 2006, 137, 1765.
Langmuir 2010, 26(2), 802–808
The final part of our investigation was focused on the applicability of the approximated approach proposed previously by us in order to describe the combined effect of the intraparticle diffusion and surface reaction kinetic mechanisms. We have shown there that the kinetics of dyes sorption may be described as a two-step process. The first (initial) kinetics is controlled by the rate of surface reaction, expressed in terms of SRT approach. Further, when the adsorbed amount reaches a certain value, switching takes place to another kinetics governed by the rate of intraparticle diffusion (described by the IDM approach). Here, we have tested if this approximate method is able to approximate well the results obtained by using the “exact” approach, i.e., by the direct combination of eqs 16 and 17 and 18-20. The equations developed recently by us were modified to the desired form by using the set of variables 15 and are used to calculate the whole composite kinetic isotherm. Details related to the numerical procedure, applied previously for the analysis of experimental data, can be found in ref 20. Here, the Q(τ) data generated by using eqs 16 and 19 were analyzed by this approximate method (see Figure 7). The results of this comparative analysis can be summarized as follows: 1 According to the approximate method, the influence of the surface reaction mechanism is dominating when τ < 16. For higher times, the overall sorption rate can be considered as controlled mostly by the intraparticle diffusion. 2 The qualitative differences between the kinetic isotherm calculated by eqs 16 and 19 and the “composite” Q(τ) function are not significant (this includes the concavity of Q(τ1/2) isotherm). Thus, the method described in ref 20 is able to fit the real experimental data equally well in comparison to the exact approach. 3 Both methods differ essentially when considering the obtained values of coefficients (treated as the best-fit parameters). The obtained values of φ1 are significantly lower than the actual ones (up to 50%). The same can be said about the ζ value, but now the adjusted value can be higher than the correct one up to 1 order of magnitude. Thus, the approximated method can be successfully used to fit data, but the obtained values of best-fit parameters can not be treated as having solid physical meaning.
5. Conclusions 1 The novel two-resistance model was developed for theoretical description of the sorption kinetics. Both DOI: 10.1021/la902211c
807
Article
the intraparticle diffusion and the surface reaction kinetic steps were taken into account. Two kinds of sorbent particle geometry were considered: spherical and plane particles. The rate of surface reaction was described in terms of the new SRT (Statistical Rate Theory) approach and the first-order reversible kinetics. 2 Model calculations allowed for determining the coefficient values (associated with the “rate constant” term) above and below which the behavior of the system can be described in terms of single kinetic step. Similar analysis may simplify the theoretical description of a given sorption system.
808 DOI: 10.1021/la902211c
Plazinski and Rudzinski
3
4
The concave character of the Q(τ1/2) kinetic isotherms was observed when the two-resistance model was applied. This observation, combined with the results of our previous study 19 let to state that such behavior is of a general nature and is connected with the situation when at least two different processes are involved in controlling the sorption kinetics. A brief discussion on both equilibration times and the applicability of the approximate method of analysis of experimental data (proposed originally in one of our previous papers)20 was presented.
Acknowledgment. W.P. acknowledges the financial support of the Foundation for Polish Science (START program, 2009).
Langmuir 2010, 26(2), 802–808
