Models for adsorption kinetics

The simplest kinetic model for adsorption is that proposed by Langmuir [422], in which it is assumed that (i) the surface is homogeneous (ii) every 

Expressed in terms of the sticking probability, the Langmuir expression is simply 

Two distinct approaches have been used to model precursor state kinetics. (1) A successive site statistical model, introduced by Kisliuk [426] for adsorption and adapted by King [298] for desorption. (2) The chemical reaction kinetics approach, involving rate coefficients and the stationary state approximation, followed by Becker and Hartman [424], Ehrlich [425] and recently developed by Gorte and Schmidt [297] and Cassuto and King [421], It has recently been shown by Schon-hammer [427] and Cassuto and King [421] that the two approaches produce the same kinetic expressions. Variants of these models have 

kd and km are the rate coefficients for adsorption, desorption, and migration from the intrinsic precursor state, k m and are the rate coefficients for migration and desorption from the extrinsic precursor state, kv is the rate coefficient for transfer from the chemisorbed state to the intrinsic precursor state, and a and a are the trapping probabilities for molecules incident at intrinsic and extrinsic precursor sites, respectively. Direct transfer from gas phase to chemisorbed state or vice versa is included through the probability, sc, for adsorption and the rate coefficient, ftc, for desorption [427]. In order to generalise the rate expressions, we now introduce a group of terms F(0) which are only functions of the surface coverage 0. For a particular case, such as non-dissociative adsorption, these terms may be evaluated and inserted into the appropriate rate expression. 

Fd is the occupation probability that a site (or sites) exists in a configuration which can lead to desorption (= 0 for non-dissociative adsorption). 

---

Winters and Lee134 describe a physically based model for adsorption kinetics for hydrophobic organic chemicals to and from suspended sediment and soil particles. The model requires determination of a single effective dififusivity parameter, which is predictable from compound solution diffusivity, the octanol-water partition coefficient, and the adsorbent organic content, density, and porosity. 

On the basis of a molecular model for adsorption kinetics Jantti [1,2] introduced a method for the fast calculation of adsorption equilibria. He measured the actual adsorbed amount at three times shortly after a change of the sorptive gas pressure. The method was designed for use in the case of an adsorption satisfying the equation... 

The first physically sound model for adsorption kinetics, which was derived by Ward and Tordai [18], is based on the assumption that the time dependence of a surface or interfacial tension (which is directly proportional to the surface excess F, in mol m ) is caused by diffusion and transport of surfactant molecules to the interface. This is referred to as diffusion-controlled adsorption kinetics model . The interfacial surfactant concentration at any time t, T(t), is given by the following expression,... 

In a recent paper, Seri-Levy and Avnir [5] propose a model for adsorption kinetics in diffusion-limited conditions. This model is based on an extension of Delahay s analysis... 

In this section we discuss in some detail a macroscopic approximate model for adsorption kinetics on fractal interfaces. Further details can be found in [8. In diffusion-limited conditions, the balance equations for adsorption on flat surfaces take the form... 

General models for adsorption kinetics and relaxations of surfactants... 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

Faghoni F, Goddard WA. 2005. Energetics of hydrogen coverage on group VIII transition metal surfaces and a kinetic model for adsorption/desorption. J Chem Phys 122 014704. 

Adsorption has a significant impact on the movement of allelo-chemical substances in soil. Such movement in soil by water is important from the standpoint of mechanism of phytotoxin activity in the receiving species at a site remote from the donor plant. Adsorption reduces the solute concentration in the soil solution and consequently minimizes redistribution in the environment. Solute transport has been described by Pick s second law of diffusion and the kinetic models for adsorption and degradation of reactive solutes (, 44). The contribution of adsorption is measured and expressed as the retardation factor, R. 

Lindstrom, F. T., Haque, R., and Coshow, W. R. (1970). Adsorption from solution. III. A new model for the kinetics of adsorption-desorption processes. J. Phys. Chem. 74, 495-502. 

S. Farooq M. N. Rathor K. Hidajat. A predictive model for a kinetically controlled pressure swing adsorption separation process. Chem. Eng. Sci. 1993,48,4129. 

Farooq, S. Rathor, M.N. Hidajat, K. A Predictive Model for a Kinetically Controlled Pressure Swing Adsorption Separation Process. Chenu Eng. ScL 1993,48,4129. 

Sircar S. and Hufion J. R., Why Does die Linear Drving Force Model for Adsorptio Kinetics Work , Adsorption, 6 (2000), pp. 137-147. 

K. Schdnhammer. On the Kisliuk Model for Adsorption and Desorption Kinetics. Surf Sci. 83-.L633 (1979). 

According to the kinetic model for adsorption, the surface of the solid is assumed to serve as effective adsorption sites [39, 40]. When the rate of condensation of the gas molecules on the unoccupied sites on the surface is equal to the rate of evaporation of the gas molecules from the occupied sites, a state of dynamic equilibrium is reached, which leads to the following equation ... 

Theoretical models have reached a state that allows a quantitative description of the equilibrium state by thermodynamic models, the adsorption kinetics of surfactants at fluid interfaces, the transfer across interfaces and the response to transient or harmonic perturbations. As result adsorption mechanisms, exchange of matter mechanisms and the dilational rheology are obtained. For some selected surfactant systems, the characteristic parameters obtained on the various levels coincide very well so that a comprehensive understanding was reached. 

The modelling of adsorption kinetics of surfactants, which show interfacial aggregation has been discussed recently elsewhere (Fainerman et al. 2000, 2000a Aksenenko 1998). To derive an adsorption kinetics model the Ward and Tordai equation (29) is again the main relationship between the dynamic adsorption F(t) and the subsurface concentration c(0,t), and as adsorption isotherm the equations (21) - (23). The set of equation is too complex to find an analytical solution, but for the short time range and for low adsorption layer coverage, the following approximation is valid (Aksenenko et al. 1998)... 

Nagatani, H., D.J. Fermin, and H.H. Girault (2001). A kinetic model for adsorption and transfer of ionic species at polarized liquid/liquid interfaces as studied by potential modulated fluorescence spectroscopy. J. Phys. Chem. B 105, 9463-9473. 

Previonsly the inflnence of adsorbed material on contact angles has been seen to be important. In general, the stndy of adsorption has been prompted by its importance in catalysis and reactions at interfaces. The Langmnir-Hinshelwood model for the kinetics of snch reactions is popnlar and depends on the adsorption-desorption rates (Bondart and Djega-Mariadasson, 1984). The nature of adsorption also needs to be stndied to determine the viability of chromatographic separation processes for a given system. Its attraction is that it is one of the very few means for separating similar components, snch as isomers, azeotropes, etc., that are traditionally difficnlt (Holland and Liapis, 1983). 

---