Linear adsorption limit

A (Figure 4.9). The diameter of such a neck, 2.3 A, is sufficiently large for a linear C-C chain to pass, but too small to also be an equilibrium adsorption position. The largest compound allowed inside the pores is a linear molecule limited in length to four carbon atoms due to the distance between two subsequent necks [103]. Another example of shape-selective behavior is found in a Zn-based MOF able to encapsulate linear hexane while branched hexanes are blocked [104]. 

The main limitation of the Freundhch equation is that it does not predict a maximum adsorption capacity, because linear adsorption generally occurs at very low solute concentration and low loading of the sorbent. However, in spite of this limitation, the Freundlich equation is used widely for describing contaminant adsorption on geosorbents. 

Traditional (i.e., non-MIP) SPE sorbents are similar to HPLC stationary phases. The advantages of many of these materials are that they are widely available, well characterized, have high binding capacity, and show linear adsorption behavior. One may observe that just a few types are used in the majority of sample preparations, i.e., these materials are quite generic, and it is the wash and elution step which is varied according to the application. The generic nature of these materials is also a drawback because it reflects their limited selectivity. 

As columns are overloaded for preparative work, peak shape often deviates from the Gaussian shape typical of analytical work. In preparative work, the peaks can assume a triangular shape because the adsorption isotherm is nonlinear. A typical isotherm is shown in Figure 6-37, where CM is the concentration of sample in the mobile phase and Cs is the concentration of sample in the stationary phase. At low concentration of sample (CM) there is a linear adsorption isotherm which results in Gaussian peak shapes. At a point when either the sample adsorption in the stationary phase or the sample solubility in the mobile phase becomes limited, the isotherm becomes nonlinear, assuming either a convex or a concave shape. Convex isotherms are the most common and result in peak tailing. Conversely, concave isotherms cause fronting of the peaks. 

If the cumene decompostion is adsorption limited, then the initial rate will be linear with the initial partial pressure of cumene as shown in Figure 10-12. 

Most experimental adsorption isotherms can be considered as a combination of these two ideal types. For heterogeneous surfaces, adsorption isotherms are often modeled as a combination of L and S adsorption isotherms corresponding to a distribution of patches [49, 50]. The many other shapes proposed in the preceding classifications [48] Hke stepwise, high afiinity, or linear can be considered either as the combinations of S- and L-types or as a representation of the phenomenon for a limited range of concentration. For example, the high-affinity type is an extreme form of L-type. A linear adsorption isotherm (if it is not an artefact due to the penetration of the solute in the solid [15] may be the first portion of an L-type observed in the low concentration range. 

A simple equation derived by Sutherland (1952) can be used as a first approximation but its range of application is very limited (Miller 1990). The derivation of Sutherland is based on a linear adsorption equation of the form... 

Fig. 4 Non-retained (or split-peak) fractions observed for injections of rabbit immunoglobulin G onto an immobilized protein G column at various flow rates. Source From Non-linear elution effects in split-peak chromatography. II. Role of ligand heterogeneity in solute binding to columns with adsorption-limited kinetics, in J. Chromatogr.

In EIS one can use potential or current sinusoidal perturbations. In practice, the potential perturbation of 10 mV peak to peak or a 5 mV amphtude is usually used because EIS is based on the linearization of nonlinear electrochemical equations. This also means that as the sum of sine waves is appUed, its total amplitude cannot exceed 5 mV. In practice amplitude of 5 mV rms is usually used for diffusion and adsorption limited processes, see Sect. 13.2, but in certain cases of surface processes where sharp voltammetric peaks appear the amplitude should be much lower. The linearity can be simply checked by decreasing amplitude and comparing the obtained results. Sect. 13.2. It should be kept in mind that the apparatus used in electrochemistry displays the root-mean-squared (rms) amplitude, which is the effective amplitude measured by an ac voltmeter. This rms amplitude is equal to the real amplitude divided by V ... 

As the Ward-Tordai equation contains two independent variable functions (surface excess and subsurface concentration), its application requires a further equation relating the two functions. The first attempt at this was by Sutherland [11], who incorporated a linear adsorption isotherm. This, however, proved to be quite limiting, and so various other isotherms were employed [12, 13]. Even so, these extended theorems accurately matched experimental results only in the case of some nonionic surfactants. 

In this review, we have summarized theoretical concepts and recent advances for the adsorption of linear polyelectrolyte molecules onto curved surfaces in the weak and strong adsorption limit. A mean-field description is adopted, and the interaction potentials between the polyelectrolyte and the surfaces are derived from the linearized Poisson-Boltzmann equation for the corresponding geometries (planes, cylinders, and spheres). The derivation of an exact analytical solution of the adsorption problem for curved surfaces is a major challenge and is yet unsolved. 

The first evidenee of the existence of an end is the so-called limiting state of the supercritical adsorbate defined by the intersection of linear isotherms [80]. Such linear adsorption isotherms of CH4 on activated carbon are shown for the region of low surface concentration in Fig. 14. All isotherms in the supercritieal region, except the one at 198 K, intersect each other at one point. This point defines a target of absolute amount in the adsorbed phase per gram of adsorbent that can possibly be reached following any course of pressure increment at any temperature. Therefore, it was defined previously as limiting adsorption,... 

Many experimental works have been reported recently dealing with kinetic aspect of colloid and protein adsorption at solid-hquid interfaces. These results have been reviewed in some detail elsewhere [1,2,7,12,14,76,116]. In this section, we present some representative experimental results obtained under well-defined transport eonditions whieh eonfirm the validity of the theoretical approaches discussed earlier. The usefiilness of the eoUoid system to mimie the adsorption processes of molecules also will be pointed out. The data eonceming limiting flux measurements (linear adsorption regime) are discussed first, whereas the last part of this section will be focused on describing the effect of surface-blocking effects (steric barrier). 

FIG. 16-2 Limiting fixed-bed behavior simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodtigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dotdtecht, The Nethedands, 1989 reptinted withpeimission.]... 

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