INTRODUCTION:
Adsorption is the adhesion of atoms, ions, biomolecules or molecules of gas, liquid, or dissolved solids to a surface. Adsorption is present in many natural physical, biological, and chemical systems, and is widely used in industrial applications such as activated charcoal for capturing and using waste heat to provide cold water for air conditioning and other process requirements (adsorption chillers), synthetic resins, increase storage capacity of carbide-derived carbons for tunable nanoporous carbon, and water purification. Adsorption, ion exchange, and chromatography are sorption processes in which certain adsorbates are selectively transferred from the fluid phase to the surface of insoluble, rigid or packed particles suspended in a column. Lesser known, are the pharmaceutical industry applications as a means to prolong neurological exposure to specific drugs or parts thereof. Adsorption is one of the most widely applied techniques for pollutant removal from contaminated medias. The common adsorbents include activated carbon, molecular sieves, polymeric adsorbents, and some other low-cost materials. When adsorption is concerned, thermodynamic and kinetic aspects should be involved to know more details about its performance and mechanisms. Except for adsorption capacity, kinetic performance of a given adsorbent is also of great significance for the pilot application.
From the kinetic analysis, the solute uptake rate, which determines the residence time required for completion of adsorption reaction, may be established. Also, one can know the scale of an adsorption apparatus based on the kinetic information. Generally speaking, adsorption kinetics is the base to determine the performance of fixed-bed or any other flow-through systems.
In the past decades, several mathematical models have been proposed to describe adsorption data, which can generally be classified as adsorption reaction models and adsorption diffusion models. Both models are applied to describe the kinetic process of adsorption; however, they are quite different in nature. Adsorption diffusion models are always constructed on the basis of three consecutive steps (Lazaridis and Asouhidou, 2003): (1) diffusion across the liquid film surrounding the adsorbent particles, i.e., external diffusion or film diffusion; (2) diffusion in the liquid contained in the pores and/or along the pore walls, which is so-called internal diffusion or intra-particle diffusion; and (3) adsorption and desorption between the adsorbate and active sites, i.e., mass action. However, adsorption reaction models originating from chemical reaction kinetics are based on the whole process of adsorption without considering these steps mentioned above.
At present, adsorption reaction models have been widely developed or employed to describe the kinetic process of adsorption 1-12; however, there still exist some problems. For example, pseudo-second-order rate equation based on chemical adsorption was unsuitably employed to describe organic pollutants adsorption onto several non-polar polymeric adsorbents. This is essentially a process of physical adsorption13. In addition, Lagergren 14 models were still widely applied to data modeling, though no adsorption mechanisms could be reasonably available. Consequently, some famous journals like Journal of Hazardous Materials and Separation and Purification Technology began to decline adsorption manuscripts based on unsuitable or simple kinetic modeling as mentioned above15and16.
Here we reviewed several widely-used batch kinetic models and paid more attention to their boundary conditions. We believe the subject review is of considerable significance to improve the current research on adsorption kinetic modeling.
ADSORPTION REACTION MODELS:
Pseudo-first-order rate equation:
Lagergren14 presented a first-order rate equation to describe the kinetic process of liquid-solid phase adsorption of oxalic acid and malonic acid onto charcoal, which is believed to be the earliest model pertaining to the adsorption rate based on the adsorption capacity. It can be presented as follows:
--------- 1
Where, qe and qt (mg/g) are the adsorption capacities at equilibrium and time t (min), respectively. kp1 (min−1) is the pseudo-first- order rate constant for the kinetic model. Integrating Eq.(1) with the boundary conditions of qt=0 at t=0 and qt=qt at t=t, yields17 .
--------- 2
Which can be rearranged to:
---------- 3
To distinguish kinetic equations based on adsorption capacity from solution concentration, Lagergren’s first order rate equation has been called pseudo-first-order18. In recent years, it has been widely used to describe the adsorption of pollutants from wastewater in different fields, such as the adsorption of methylene blue from aqueous solution by broad bean peels and the removal of malachite green from aqueous solutions using oil palm trunk fibre19and 20.
Pseudo-second-order rate equation:
In 1995, Ho described a kinetic process of the adsorption of divalent metal ions onto peat21, in which the chemical bonding among divalent metal ions and polar functional groups on peat, such as aldehydes, ketones, acids, and phenolics are responsible for the cation-exchange capacity of the peat. Therefore, the peat-metal reaction may be presented as shown in Eqs.(4) and (5), which can be dominant in the adsorption of Cu2+ ions onto peat22:
2P- + Cu2+ ↔ CuP2 and
2HP + Cu2+ ↔ CuP2 + 2H+
Where, P- and HP are active sites on the peat surface. The main assumptions for the above two equations were that the adsorption may be second-order, and the rate limiting step may be chemical adsorption involving valent forces through sharing or the exchange of electrons between the peat and divalent metal ions. In addition, the adsorption follows the Longmuir equation23.
The rate of adsorption described by Eqs.(4) and (5) is dependent upon the amount of divalent metal ions on the surface of peat at time t and that adsorbed at equilibrium. Therefore the rate expression may be given as
Or
Where, (P)0 and (HP)0 denote the amount of equilibrium sites available on the peat, (P)t and (HP)t denote the amount of active sites occupied on the peat at time t, and kp2 (g/(mg·min)) is the pseudo-second-order rate constant of adsorption21 . The driving force, (qe−qt), is proportional to the available fraction of active sites24. Then, it yields
---------- 8
Eq.(8) can be rearranged as follows:
----------- 9
Integrating Eq.(9) with the boundary conditions of qt=0 at t=0 and qt=qt at t=t, yields
---------- 10
Which can be rearranged as follows:
--------- 11
and
-----------12
Where, V0 (mg/(g·min)) means the initial adsorption rate, and the constants can be determined experimentally by plotting of t/qt against t.
Similarly, Ho’s second-order rate equation has been called pseudo-second-order rate equation to distinguish kinetic equations based on adsorption capacity from concentration of solution24. This equation has been successfully applied to the adsorption of metal ions, dyes, herbicides, oils, and organic substances from aqueous solutions 25-29.
Elovich’s equation:
A kinetic equation of chemisorption was established by Zeldowitsch30 and was used to describe the rate of adsorption of carbon monoxide on manganese dioxide that decreases exponentially with an increase in the amount of gas adsorbed24, which is the so-called Elovich equation as follows31:
---------------- 13
Where, q represents the amount of gas adsorbed at time t, a the desorption constant, and α the initial adsorption rate18. Eq.(13) can be rearranged to a linear form:
-----14
with
The plot of q versus log (t+t0) should yield a straight line with an opposite value of t0. Elovich equation is applied to determine the kinetics of chemisorption of gases onto heterogeneous solids32. With the assumption of αt>>133, Eq.(14) was integrated by using the boundary conditions of q=0 at t=0 and q=q at t=t to yield 24:
q = α ln(aα) +α lnt ----------------- 16
Elovich’s equation has been widely used to describe the adsorption of gas onto solid systems34. Recently it has also been applied to describe the adsorption process of pollutants from aqueous solutions, such as cadmium removal from effluents using bone, and Cr(VI) and Cu(II) adsorption by chitin, chitosan, and Rhizopus arrhizus 36 .
Second-order rate equation:
The typical second-order rate equation in solution systems is
-------------------
17
Eq.(17) was integrated with the boundary conditions of Ct=0 at t=0 and Ct=Ct at t=t to yield
---------------------- 18
Where, C0 and Ct (mg/L) is the concentration of solute at equilibrium and at time t (min), respectively, and k2 (L/(mg·min)) is the rate constant.
In earlier years, the second-order rate equations were reasonably applied to describe adsorption reactions occurring in soil and soil minerals. Recently, the equation has also been used to describe fluoride adsorption onto acid-treated spent bleaching earth and phosphamidon adsorption on an antimony (V) phosphate cation exchanger37.
ADSORPTION DIFFUSION MODELS:
It is generally known that a typical liquid/solid adsorption involves film diffusion, intraparticle diffusion, and mass action. For physical adsorption, mass action is a very rapid process and can be negligible for kinetic study. Thus, the kinetic process of adsorption is always controlled by liquid film diffusion or intrapartical diffusion, i.e., one of the processes should be the rate limiting step . Therefore, adsorption diffusion models are mainly constructed to describe the process of film diffusion and/or intrapartical diffusion.
Liquid film diffusion model:
1. Linear driving force rate law
In liquid/solid adsorption systems the rate
of solute accumulation in the solid phase is equal to that of solute transfer
across the liquid film according to the mass balance law. The rate of solute
accumulation in a solid particle clearly equals to
Where q represents the average solute concentration in the solid, and Vp the volume of the particle. Meanwhile the rate of solute transfer across the liquid film is proportional to the surface area of the particle As and the concentration driving force (C−Ci). Therefore, it equals to kfAs(C−Ci), where kf represents the film mass transfer coefficient. With the discussion above we can obtain
----------------
19
where Ci and C denote the concentration of solute at the particle/liquid interface and in the bulk of the liquid far from the surface, respectively. Eq.(19), thus, can be rearranged to
-----------------------
20
The ratio As/Vp that is the particle surface area per unit particle volume can be defined as S0. Then, Eq.(20) can be written as
---------------------- 21
Eq.(21) is called as “linear driving force” rate law, which is usually applied to describe the mass transfer through the liquid film.
2. Film diffusion mass transfer rate equation
The film diffusion mass transfer rate equation presented by Boyd38 is
…………………………22
…………………………….23
where Rl (min−1) is liquid film diffusion constant, Del (cm2/min) is effective liquid film diffusion coefficient, r0 (cm) is radius of adsorbent beads, Δr0 (cm) is the thickness of liquid film, and k′ is equilibrium constant of adsorption.
A plot of ln(1−qt/qe)~t should be a straight line with a slope −Rl if the film diffusion is the rate limiting step. Then the corrected liquid film diffusion coefficient Del can be evaluated according to Eq.(21). The film diffusion mass transfer rate equation has been successfully applied to model several liquid/ solid adsorption cases, e.g., phenol adsorption by a polymeric adsorbent NDA-100 under different temperature and initial concentration conditions13.
Intraparticle diffusion model:
1. Homogeneous solid diffusion model (HSDM)
A typical intraparticle diffusion model is the so-called homogeneous solid diffusion model (HSDM), which can describe mass transfer in an amorphous and homogeneous sphere. The HSDM equation can be presented as
-------------------- 24
Where, Ds is intraparticle diffusion coefficient, r radial position, and q the adsorption quantity of solute in the solid varying with radial position at time t. Crank39 gave an exact solution to Eq.(24) for the “infinite bath” case where the sphere is initially free of solute and the concentration of the solute at the surface remains constant. External film resistance can be neglected according to the constant surface concentration. Then, Crank’s solution is written as follows
------- 25
Where, R is the total particle radius. The average value of q in a spherical particle at any particular time, defined as q, is presented as follows:
-------------------- 26
Where, q(r) is the local value of the solid-phase concentration. By inserting the solution for q(r) into Eq.(26), the following equation can be obtained:
-------------- 27
Where, q∞ represents the average concentration in the solid at infinite time. For a short time, when q/q 0.3, ∞ < Eq.(27) can be simplified to yield
------------------- 28
A value of Ds from short-time data can be determined by plotting of q/q∞ against t1/ 2 . It can also be concluded from Eq.(28) that adsorption rate reduces along with increasing adsorbent particle size and vice versa.
For a long time, Eq.(27) may be written as follows:
------------------- 29
The linearization of Eq.(29) gives
------------- 30
Similarly, the value of Ds from long-time data can be determined by plotting of ln(1 q/q ) ∞ − vs t. However, the assumption of constant surface concentration for HSDM is likely to be violated at a long time. Therefore, the equation discussed above is generally somewhat valid in a short time.
In recent years, HSDM has been applied to different kinds of adsorption systems40, such as the adsorption of salicylic acid and 5-sulfosalicylic acid from aqueous solutions by hyper cross linked polymeric adsorbent NDA-99 and NDA-101. In the adsorption systems of pentachlorophenol (PCP) onto activated carbon, diffusion coefficient Ds evaluated from batch kinetic adsorption experiments has been applied to fixed-bed situation, such as the prediction of the adsorption breakthrough curves and design of fixed beds for removal of PCP 41.
2. Weber-Morris model
Weber-Morris found that in many adsorption cases, solute uptake varies almost proportionally with t1/2 rather than with the contact time t42:
---------------------
31
Where, kint is the intraparticle diffusion rate constant. According to Eq.(31), a plot of qt~t1/2 should be a straight line with a slope kint when the intraparticle diffusion is a rate-limiting step. For Weber-Morris model, it is essential for the qt~t1/2 plot to go through the origin if the intraparticle diffusion is the sole rate-limiting step. However, it is not always the case and adsorption kinetics may be controlled by film diffusion and intraparticle diffusion simultaneously. Thus, the slope is not equal to zero.
3. Dumwald-Wagner model
Dumwald-Wagner proposed another intraparticle diffusion model as43
------- 32
where K (min−1) is the rate constant of adsorption. Eq.(32) can be simplified as
------------------
33
A plot of log(1−F2)~t should be linear and the rate constant K can be obtained from the slope. Dumwald-Wagner model proved to be reasonable to model different kinds of adsorption systems, e.g., p-toluidine adsorption from aqueous solutions onto hyper cross linked polymeric adsorbents.
Double-exponential model (DEM):
A double-exponential function proposed by Wilczak and Keinath44 was used to describe lead and copper adsorption onto activated carbon. In this case, the uptake process of both metals could be divided into two steps, namely a rapid phase involving external and internal diffusions, followed by a slow phase controlled by the intraparticle diffusion. It was demonstrated that the two-step mechanism can be described fairly well with the double-exponential model45, which is presented as follows:
----------- 34
Where, D1 (mmol/L) and K1 (min−1) are the adsorption rate parameters and diffusion parameters of the rapid step, respectively, and D2 and K2 for the slow step. If K1>>K2, it means that the rapid process can be assumed to be negligible on the overall kinetics.
Eq.(34) can then be simplified as
------------------
35
and be rearranged to a linear form
-----------------
36
the parameters D2 and K2 can be determined by plotting of e ln(qe-qt) against t, then D1 and K2 can be obtained from the following equation:
-------------- 37
Then, Eq.(34) can be used to fit the kinetic process of adsorption.
Nevertheless, values of K1 and K2 are not sufficient to describe and interpret the influence of external and internal diffusions. As the rapid step involves both diffusion steps, these parameters can only allow a comparison of the respective adsorption rate of Pb(II) and Cu(II).
DEM can also describe a process where the adsorbent offers two different types of adsorption sites. On the fir st-type site rapid adsorption equilibration occurs within a few minutes, whereas on the second site type, adsorption is more slowly. DEM is particularly suitable for modeling heavy metals adsorption, e.g., adsorption of Cu(II) and Pb(II) from aqueous solutions y activated carbon and grafted silica45 .
CONCLUSIONS AND PERSPECTIVES:
Both adsorption reaction models and adsorption diffusion models are now widely employed for fitting kinetic data. Based on the above analysis, it can be seen that appropriate adsorption reaction models should be chosen according to the mechanism of adsorption process and the applicability of different models. Generally speaking, they cannot represent the of adsorption and thus, cannot offer useful information to gain insight in adsorption mechanism and to design fixed-bed systems. On the contrary, adsorption diffusion models based on three basic steps can represent the real adsorption course more reasonably, and intraparticle diffusion coefficient Ds determined from these models are useful for design of fixed-bed systems.
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Received on 03.12.2011 Modified on 22.12.2011
Accepted on 30.12.2011 © AJRC All right reserved
Asian J. Research Chem. 5(1): January 2012; Page 08-13