Adsorption kinetics model

First-order adsorption kinetics model A simple first-order reaction model is based on a reversible reaction with equilibrium state being established between two phases (A— fluid, B—solid) ... 

The experimental results were modeled using the adsorption/kinetic model and the parameters listed in Table VIII. 

An adsorption kinetic model was developed to evaluate the adsorption rates of five pure gases (Nj, O2, Ar, CO, and CH4) on a Takeda-3A CMS over a wide range of pressures up to ISatm. The kinetic characteristics of adsorption on the CMS were studied by using the adsorption equilibrium of five pure gases measured at three different temperatures and their physical properties. Since the diffiisional time constants of all the components showed much stronger dependence of pressure than those expected by the traditional Darken relation, a structural diffusion model was applied to predict the strong pressure dependence. The proposed model successfully predicted the dif ional time constant up to high pressure on the CMS. 

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,... 

Values of reaction orders (Table 3) and competitive sorption effects mentioned in Section 3.3 could be quantified more satisfactorily using a adsorption kinetic model. The model was derived under the assumptions of constant volume reaction, nitrous oxide reacting from the gas phase (because of the small influence of nitrous oxide feed concentration on phenol selectivity), three times higher phenol sorption constant than benzene sorption constant (Kc6H5oh 3 Kc6H6, 3s derived from sorption simulation calculations), but without considering the dependency on temperature (reaction temperature 400°C) [5,7]. 

In order to confirm their proposed mechanism (surface reaction between chemisorbed benzene and chemisorbed hydrogen), poisoning experiments were employed using thiophene as a poison. This showed that the deactivation rate decreases with increasing hydrogen and benzene partial pressures, and hence they concluded from their data that hydrogen and benzene are both chemisorbed on the catalyst surface and their proposed dual-site adsorption kinetic model is suitable to describe the hydrogenation of benzene on nickel. 

The adsorption kinetics of interfacial active molecules at liquid interfaces, for example surfactants at the aqueous solution/air or solution/organic solvent interface, can be described by quantitative models. The first physically founded model for interfaces with time invariant area was derived by Ward Tordai (1946). It is based on the assumption that the time dependence of interfacial tension, which is directly correlated to the interfacial concentration T of the adsorbing molecules, is caused by a transport of molecules to the interface. In the absence of any external influences this transport is controlled by diffusion and the result, the so-called diffusion controlled adsorption kinetics model, has the following form... 

Adsorption Kinetics Model for the Maximum Bubble Pressure Method... 

In coating processes the problem of controlling the flow of liquids down an inclined plate is a key question (Scriven 1960, Kretzschmar 1974). Therefore, the hydrodynamic flow of such films in combination with surface rheological and adsorption kinetics models were described. As the principle of a flowing film can be used also as a separate method to study adsorption processes in the range of milliseconds, the theory is presented here, while the experimental details are given in the next chapter. 

The disadvantage of this thermodynamic criterion is that it can be used only in combination with a purification procedure. Thus, if such a procedure is not available, the purity of a surfactant solution with respect to its useful interfacial study caimot be checked. Therefore, a second criterion was developed, which is based on an adsorption kinetics model. If adsorption of both, the main component and the impurity is assumed to be diffusion-controlled, the difference between the surface tension values, measured after a definite time t j after adsorption and desorption, respectively, is a measure of the purity of the solution. 

Adsorption Kinetics Model, Taking Into Account the Electrostatic Retardation and a Specific Adsorption Barrier... 

Appendix 4E Application of the Laplace Transform to solve the diffusion-controlled ADSORPTION KINETICS MODEL... 

As mentioned above many adsorption kinetics models were discussed in the literature. These models consider specific mechanisms of the molecular transfer from the subsurface to the interface, or vice versa in the case of desorption [5, 6, 7, 8, 9, 10, 11, 12, 14, 23], The models, which assume only the transfer mechanism as rate determining step are called kinetic-controlled. More advanced models, the so-called mixed models, consider both the transport by diffusion in the bulk and the transfer mechanism (cf. Fig. 4.1). Such mixed models were first derived by Baret in 1969 [9] who combined Eq. (4.1) with a certain transfer mechanism. 

To derive an adsorption kinetics model the Ward and Tordai equation (4.1) is again the main relationship between the dynamic adsorption r(t) and the subsurface concentration c(0,t). As it was described in detail in paragraph 4.1.2, an adsorption isotherm as additional function r(c) is needed for a kinetic model. The isotherm equations (2.110) - (2.112) given in Chapter 2 represent such type of function, which accounts for a 2D-aggregation in the adsorption layer [48]. The set of equation is too complex to find an analytical solution. Only for the short time range and for low adsorption layer coverage, the following approximation is valid [65]... 

As it is the case in adsorption kinetics models, besides the diffusion theory, other models exist to describe the exchange of matter. We refer here only to the review by Miller et al. [1] and the book by Dukhin et al. [2]. 

The graph in Fig. 41 shows the dynamic surface tensions of a mixtured solution of CioDMPO and C14DMPO measured with the maximum bubble pressure method BPAl (O) and profile analysis tensiometer PATl ( ). The theoretical curves shown were calculated due to the adsorption kinetics model for surfactant mixtures discussed above (Miller et al. 2003). 

Some adsorptive kinetic models were used in different works of adsorption of organic compounds in organophilic clays, including models of Lagergren, also called models of pseudo-first order and pseudo-second order, represented by Equations 13 and 14 (Akgay, 2004, 2005 Cavalcanti et al., 2008 Yilmaz Yapar, 2004). 

Among the isotherms that portray the adsorptive equilibrium, the Langmuir isotherm, Freundlich and Dubinin-Raduskevich were cited. For the adsorptive kinetic models, stand out the Lagergren, in models of pseudo-first order and pseudo-second order. 

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]. 

The time required for surface tension reduction depends on diffusion processes involved in surfactant adsorption. Kinetic models for surfactant adsorption divide the adsorption process into two steps [67]. The first step is the transport of the surfactant to the subsurface, driven by a concentration gradient or hydrody-... 

Above expression is used solely as adsorption equilibrium formulation in adsorption kinetics modeling and not as true representation of exchange kinetics mechanism between pore fluid and adsorbed phase. 

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