Equilibrium constant for adsorption

One notes in Table 1.2 a uniform increase in the adsorption energies of the alkanes when the microspore size decreases (compare 12-ring-channel zeohte MOR with 10-ring-channel TON). However, at the temperature of hydroisomerization the equilibrium constant for adsorption is less in the narrow-pore zeohte than in the wide-pore system. This difference is due to the more limited mobility of the hydrocarbon in the narrow-pore material. This can be used to compute Eq. (1.22b) with the result that the overall hydroisomerization rate in the narrow-pore material is lower than that in the wide-pore material. This entropy-difference-dominated effect is reflected in a substantially decreased hydrocarbon concentration in the narrow-pore material. 

Note that in the final desorption step the equilibrium constant for adsorption of AB equals 1 /K4 whereas for the other adsorption steps it is defined as... 

If one takes the ratio of the pseudo rate constant for adsorption to that for desorption as an equilibrium constant for adsorption (K), equation 6.2.4 can be written as... 

For SiC>2, we have only considered sources for silica suspensions which were non-porous, such as Ludox (39), pyrogenic silica (40), heat-treated BDH silica (22), or ground quartz (41). The data from these sources at 0.1M concentration has been collected in Figure 7. The data of the various researchers is quite consistent, in spite of the differences in origin of the suspensions, and the different electrolytes used. The slope of the points above pH 7 shows that the adsorption capacitance for cations is very large for both sodium and potassium ions, around 200 pF/cm2. Such a capacitance corresponds to a distance of 0.25.X, when using the dielectric constant of immobilized water molecules. The equilibrium constant for adsorption is low, however, since both KNa+ and Kk+ lie between 0.1 and 0.01 dms/mol. A possible interpretation of these results is as follows there is little specific attraction between SiC>2 and alkali cations,... 

Co/pH and V o/pH results are sensitive to different aspects of the surface chemistry of oxides. Surface charge data allow the determination of the parameters which describe counterion complexation. Surface potential data allow the determination of the ratio /3 —< slaDL- Given assumptions about the magnitude of the site density Ns and the Stern capacitance C t, this quantity can be combined with the pHp2C to yield values of Ka and Ka2. Surface charge/pH data contain direct information about the counterion adsorption capacitances in their slope. To find the equilibrium constants for adsorption, a plot such as those in Figures 7 and 8 can be used, provided that Ka and Kai are independently known from V o/pH curves. 

In Equation 1.15, q represents the adsorbed amount of solute, ns and qs are the saturation capacities (number of accessible binding sites) for site 1 (nonstereoselect-ive, subscript ns) and site 2 (stereoselective, subscript s), and fens and bs are the equilibrium constants for adsorption at the respective sites [54]. It is obvious that only the second term in this equation is supposed to be different for two enantiomers. Expressed in terms of linear chromatography conditions (under infinite dilution where the retention factor is independent of the loaded amount of solute) it follows that the retention factor k is composed of at least two distinct major binding increments corresponding to nonstereoselective and stereoselective sites according to the following... 

The chromatographic retention factor, k, is related to the equilibrium constant for adsorption to a solid phase or distribution between liquid mobile and stationary phases by the well-known expression... 

Basic chromatographic theory relates the retention factor of a solute to the equilibrium constant for adsorption, K, according to... 

The standard deviation has been determined as ct = j where v is the number of degrees of freedom in the fit. The parameters for the molecular interaction /3, the maximum adsorption Too, the equilibrium constant for adsorption of surfactant ions Ki, and the equilibrium constant for adsorption of counterions K2, are thus obtained. The non-linear equations for the Frumkin adsorption isotherm have been numerically solved by the bisection method. 

For enzyme reactions K is the traditional Michaelis constant. For a heterogeneous surface on which adsorption and desorption but no reaction occurs (k2 = 0) K is simply an equilibrium constant for adsorption. (Actually we are not being as economical as we could be in this non-dimensionalization of the equations. We could have divided throughout by K instead of introducing the pressure scale p°, and eqn (12.9) would then have read... 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

The first application of hierarchical SA for parameter estimation included refinement of the pre-exponentials in a surface kinetics mechanism of CO oxidation on Pt (a lattice KMC model with parameters) (Raimondeau et al., 2003). A second example entailed parameter estimation of a dual site 3D lattice KMC model for the benzene/faujasite zeolite system where benzene-benzene interactions, equilibrium constants for adsorption/desorption of benzene on different types of sites, and diffusion parameters of benzene (a total of 15 parameters) were determined (Snyder and Vlachos, 2004). While this approach appears promising, the development of accurate but inexpensive surfaces (reduced models) deserves further attention to fully understand its success and limitation. 

Westall, J. and Hohl, H. A general method for the computation of equilibria and determination of equilibrium constants for adsorption at hydrous oxide surfaces, iji "Abstracts of Papers," Amer. Chem. Soc. Meeting, Miami Beach, FL., 1978. 

The equilibrium constant for adsorption of As(ni) was adjusted to give the best match to As(lll) concentrations in leachate from core 1 (Fig. 9a). The geochemical model predicted that as O2 moved through the cores and conditions became oxidizing, As(lll) should be oxidized to As(V) and strongly adsorbed, thus resulting in As concentrations below the detection limit at the pore volume when O2 broke through. 

It might be supposed that rate and equilibrium constants for adsorption or desorption could be established from pure-component adsorption data on the components involved. However, such results rarely agree with the values of the constants determined from rate data for the reaction. Interaction effects between components, other inadequacies of the Langmuir theory, and the assumption of a single controlling step, explain the deviation. 

A linear isotherm is assumed and the equilibrium constant for adsorption-desorption is assumed independent of temperature. 

The rate constant for adsorption k, surface reaction rate constant k and equilibrium constant for adsorption are all temperature dependent according to the Arrhenius expressions,... 

In triple layer approximations the location of each adsorbate with respect to the surface must be specified. Protons and all ions assumed bound as inner-sphere complexes (specifically or chemically adsorbed species) are assumed to lose part of their hydration sheaths, bonding directly to sites in the surface itself. Adsorbates assumed to remain hydrated, forming outer-sphere surface complexes, are assigned to the OHP. In the intrinsic equilibrium constants for adsorption reactions, K ", the activities of ions transferred from solution to the surface are corrected for the electrical potential they experience, % or (27). 

To simulate coadsorption of calcium and surfactant, it is first necessary to define reactions and equilibrium constants for adsorption of calcium on quartz in the absence of surfactants. The same two reactions used by other investigators to describe adsorption of divalent cations (38) were adequate to match the limited Ca adsorption data (Figure 5) published by Cooke and Digre (39). The reactions and equilibrium constants used in addition to those listed in Table I were ... 

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