Linear adsorption isotherm, assumption model

In this chapter a lumped dynamic model of a porous catalyst pellet is developed on the basis of the active site theory and assuming equilibrium adsorption-desorption according to a linear Langmuir isotherm. This model is compared with a previous pseudo-homogeneous model due to Liu and Amundson (1962). Next, the assumption of equilibrium adsorption-desorption is relaxed and the effect of both activated as well as non-activated adsorption is presented. The rate of adsorption is treated in very simple terms under the Langmuir postulates as discussed earlier. 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

The relationships of Equations 5 and 2 are unquestionably valid for unlimited surface coverage on ideal external open (flat, planar, accessible) surfaces ranging from nil at E to infinity at E=0. All of the inherent assumptions (tabulated above) are equally valid as models for physical adsorption in internal constricted regions. These are classically denoted as ultramicropores ( 2 nm), micropores(<2 nm), mesopores (2<1000nm) and macropores (very large and difficult to define with adsorption isotherm). In these instances there are finite concentration limits corresponding to the volume (space, void) size domain(s). Although caution is needed to deduce models from thermodynamic data, we can expect to observe linear relationships over the respective domains. The results will be consistent with, albeit not absolute proof of the models. 

From the above equations, models of different degrees of sophistication can be obtained. The simplest physically realistic one is that based on the assumption of equilibrium adsorption-desorption with linear Langmuir adsorption isotherm, which will be presented first, followed by the general non-equilibrium adsorption-desorption model. 

Interestingly, while the adsorption of polyelectrolytes onto BaS04 does not conform to the usual assumptions of the Langmuir adsorption isotherm model, the Langmuir model nevertheless provides a good fit to these experimental results as it does in many other cases of polymer adsorption. The Langmuir adsorption isotherm can be written in linear form as shown in Equation 1. 

Free gel was added to definite concentrations of the MB solutions (2-20 mg/1) at room temperature and was noted for its adsorption. It is clear from Figure 13.3 that the dye adsorption increases sharply with an increase in the initial dye concentration. When Cq was reached at 5 ppm and 10 ppm, the was reached at 10.04 and 20.81 respectively, which were much higher than reported Qe values of other adsorbents (Table 13.3). Equilibrium adsorption isotherm is an important criterion to determine the mechanism of dye adsorption on hydrogel. The Langmuir and Freundlich models are widely used to examine the adsorption isotherms. Freundlich isotherm models are based on the assumption that the surface of the adsorbent is not homogeneous. The experimental data in Figure 13.3 was also analyzed with the Freundlich isotherm model, which describes a heterogeneous system with multilayer adsorption. The linear form of Freundlich isotherm equation... 

Be that as it may, the adsorption isotherm has to be interpreted and one way to do this is to model the adsorption process, mathematically, in a way which contains an expression for monolayer coverage, that is, the amount of adsorbate required to cover the hypothetical surface with one layer of adsorbate. When the complexity of the adsorption process is compared with the over-simplified assumptions of the model equations to be described below, it is surprising that the equations do indeed appear to work. The reason for this is that the shape of an isotherm, quite a unique shape, is associated with distributions of adsorption potential (energy) within the porosity of the activated carbons. It is relevant to note that curves, shaped like isotherms and drawn manually without reference to adsorption data, are not linearized by adsorption equations. Draw a few curves and try this for yourself. 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

A clear weakness of the Langmuir model is the assumption that the heat of adsorption is independent of coverage. Several other isotherms have been developed which are all modifications of the Langmuir model. For example, the Temkin isotherm can be derived if a linearly declining heat of adsorption is assumed, i.e., AH = AHo(l pS), where A Ho is the initial enthalpy of adsorption. The isotherm is... 

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