Quasi-equilibrium adsorption model

Here D is the diffusion coefficient, t is the time, t is a dummy integration variable. Using Equation (8), respective T(t) dependencies can be obtained, while the Equations (l)-(7) serve as boundary condition for the diffusion model. This set of equations yield a quasi-equilibrium diffusion model which means that at a given surface pressure the composition of the surface layer under dynamic conditions is equal to that in the equilibrium. Another regime of adsorption kinetics, called kinetic model, can also be described by assuming compositions of the adsorption layer that can differ from the equilibrium state. The deviation of the adsorption layer from the equilibrium composition is the result of the finite rate of the transition process between the adsorption states. In case of two adsorption states we have6... 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

To examine the dynamical aspect of the hysteretic behavior, we consider the system geometry shown in Fig. 4. The porous material of length L in the z-direction is bounded by the gas reservoirs at z = 0 and z = L. Periodic boundary conditions are imposed on the X and y-directions. In typical experimental situations, starting from a (quasi)-equilibrium state, the external vapor pressure of the gas reservoir is instantaneously changed by a small amount, which induces gradual relaxations of the system into a new state. This geometry was used in recent work on dynamics of off-lattice models of adsorption (Sarkisov and Monson,... 

The assumption of adsorption quasi-equilibrium predicts responses which are too rapid, indicating that the sorption dynamics of the compounds is of crucial importance. When the kinetics of adsorption is included in the model, the fit of the model is significantly improved (Figure 8.14). 

Assuming that at t 00, the derivatives d/dt -> 0, one can see that the two cases defined above (quasi-equilibrium and kinetic) are described by the same boundary condition (7.47), common for all adsorption models. Then, from either Eq. (7.59) or Eq. (7.60) one obtains ... 

We have already described in Chapter V phenol photoconversion in slurry photocatalytic systems with adsoiption assumed at quasi-equilibrium. In many cases, however, photocatalytic reactors ai e operated under non-equilibrium conditions the pollutant concentrations in the solid phase and in the fluid phase are significantly fai away from the adsorption equilibrium values. As demanded by cai eful modeling, accounting for adsorption at non-equilibrium conditions is needed. 

A multicomponent model of adsorption can be reduced to three main components first, H2S, second, VOC with average properties of found species, and third, H2O. Moreover, H2O adsorption can be described as quasi-equilibrium due to the high concenbation of H2O and the long operation time of the carbon bed. Rectangular type isotherms are used to model the equilibrium of VOC and H2S. The amount of H2S adsorbed depends on available space left after VOC adsorption and it can be described bj equation ... 

The Four Kinetic Regimes of Adsorption from Micellar Solutions In the theoretical model proposed in Refs. [149,150], the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers) is avoided. The theoretical problem is reduced to a system of four nonlinear differential equations. The model has been applied to the case of surfactant adsorption at a quiescent interface [150], that is, to the relaxation of surface tension and adsorption after a small initial perturbation. The perturbations in the basic parameters of the micellar solution are defined in the following way ... 

Several theoretical models describing surface interactions between irreversibly adsorbed flexible polymers have been developed. The most common approaches are based on scaling arguments (181) or on self-consistent mean field calculations (180). The irreversibility criterion implies that the polymer adsorption/desorption rate has to be slow when compared to the approach rate of the surfaces. Under such circumstances, the total amount of polymers on the surfaces is independent of the surface separation and the system is not in true equilibrium with the bulk solution. However, it is often the case that the speed of approach is sufficiently slow for the irreversibly adsorbed polymers to adopt the most favourable conformation for each surface separation. Hence, there is equilibrium within the layer. This situation is referred to as quasi-equilibrium, or restricted equilibrium. 

This study firstly aims at understanding adsorption properties of two HSZ towards three VOC (methyl ethyl ketone, toluene, and 1,4-dioxane), through single and binary adsorption equilibrium experiments. Secondly, the Ideal Adsorbed Solution Theory (IAST) established by Myers and Prausnitz [10], is applied to predict adsorption behaviour of binary systems on quasi homogeneous adsorbents, regarding the pure component isotherms fitting models [S]. Finally, extension of adsorbed phase to real behaviour is investigated [4]. 

III. PCA was applied to eliminate some parameters from the model fitted through approach II. In consequence, it was assumed that the sites are in quasi-steady state and that AC, EC, MCC and CE are in adsorption equilibrium. It was also assumed that kinetic constants for reactions 4 and 8, and 1 and 2 are not independent and are related through a linear relation. 

The method can, however, be refined since we have made some serious approximations concerning the prefactors for the adsorption process. This is also the method typically used for microkinetic modeling of catalytic reactions. Here the dissociation of hydrogen is typically in quasi-equUibrium and, therefore, we can neglect the transition state and only concentrate on the initial and final state of the adsorption process. If we have equilibrium the chemical potentials for molecular hydrogen will be equal to the chemical potential of the two resulting adsorbed hydrogen atoms... 

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