Adsorption equilibrium, kinetics

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

If the supply of surfactant to and from the interface is very fast compared to surface convection, then adsorption equilibrium is attained along the entire bubble. In this case the bubble achieves a constant surface tension, and the formal results of Bretherton apply, only now for a bubble with an equilibrium surface excess concentration of surfactant. The net mass-transfer rate of surfactant to the interface is controlled by the slower of the adsorption-desorption kinetics and the diffusion of surfactant from the bulk solution. The characteris-... 

Kinetic Term The designation kinetic term is something of a misnomer in that it contains both rate constants and adsorption equilibrium constants. For thfe cases where surface reaction controls the overall conversion rate it is the product of the surface reaction rate constant for the forward reaction and the adsorption equilibrium constants for the reactant surface species participating in the reaction. When adsorption or desorption of a reactant or product species is the rate limiting step, it will involve other factors. 

Here we shall limit ourselves to the consideration of the influence of illumination on the adsorption equilibrium. The question as to the influence of illumination on the kinetics of adsorption will be left out. Also we shall... 

Formation of products in paraffin cracking reactions over acidic zeolites can proceed via both unimolecular and bimolecular pathways [4], Based on the analysis of the kinetic rate equations it was suggested that the intrinsic acidity shows better correlation with the intrinsic rate constant (kinl) of the unimolecular hexane cracking than with the apparent rate constant (kapp= k K, where K is the constant of adsorption equilibrium). In... 

The assessment of reaction kinetics by means of batch tests may be strongly affected by dye adsorption on the biophase and supports. The relevance of the adsorption phenomena of dyes on biophase has been addressed in studies regarding free cells [41], granular support biofilm [24], entrapped cells [11, 18], anaerobic sludge [10,24,31,34] and biological activated carbon (BAC) [42,45,47,48]. They have pointed out that the kinetics may be overestimated if the assessment of the adsorption contribution to the dye removal is not taken into account. Under batch conditions, the dye is fastly split between the liquid phase and the biophase, resulting in a sharp reduction of the dye concentration in the liquid phase until adsorption equilibrium is approached. The rate of dye adsorption must be estimated and ruled out in the kinetic assessment. 

In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the Cj M or rMI (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free c M. Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter ( ligand exchange ) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion coefficients (perhaps arising from D due to the labile complexation of M with a large macromolecule or a colloid particle, see Section 3.3). 

Wauchope and Myers [116] studied the adsorption-dispersion kinetics of Atrazine and Linuron in sediment-aqueous slurries. The resulting adsorption or desorption was very rapid, approaching 75% of equilibrium values within 3-6min. Chlorinated adsorption of the herbicide on the sediment was completely reversible after 2h of adsorption. 

Kinetically, the adsorption of humic acids at a solid-water interface is controlled by convection or diffusion to the surface. Even at concentrations as low as 0.1 mg/e near-adsorption equilibrium is attained within 30 minutes. At high surface densities, a relatively slow rearrangement of the adsorbed molecules may cause a slow attainment of an ultimate equilibrium (Ochs, Cosovic and Stumm, in preparation). The humic acids adsorbed to the particles modify the chemical properties of their surfaces, especially their affinities for metal ions (Grauer, 1989). 

Although radioactive isotopes have been widely utilized as tracers in the study of adsorption equilibrium and kinetics, in these types of studies they provide no direct information on chemical structure... 

R. M. Zimmermann, C. F. Schmidt, and H. E. Gaub, Absolute quantities and equilibrium kinetics of macromolecular adsorption measured by fluorescence photobleaching in total internal reflection, J. Colloid Interface Sci. 139, 268-280 (1990). 

The kinetic studies of the hydrogenolysis of DPM indicate that both the DPM and hydrogen are adsorbed on the same kind of active sites on the catalyst. Also, the rate-determining step of the hydrogenolysis is a surface reaction between adsorbed DPM and dissociatively adsorbed hydrogen. When the rate equation for DPM is applied to asym DAMs, their reactivities can be satisfactorily explained, and it is suggested that the product selectivity is proportional to the ratio of the adsorption equilibrium constants of the two aryl groups. 

Based on the Langmuir-Hinshelwood expression derived for a unimolecular reaction system (6) Rate =k Ks (substrate) /[I + Ks (substrate)], Table 3 shows boththe apparent kinetic rate and the substrate concentration were used to fit against the model. Results show that the initial rate is zero-order in substrate and first order in hydrogen concentration. In the case of the Schiff s base hydrogenation, limited aldehyde adsorption on the surface was assumed in this analysis. Table 3 shows a comparison of the adsorption equilibrium and the rate constant used for evaluating the catalytic surface. 

Calculated adsorption equilibrium constants indicate the Schiffs base is adsorbed more favorably on the catalyst surface than the aldehyde. This observation is consistent with situation kinetics occurring during the initial stage of the hydrogenation. The apparent rate constant shows that the product C formation is much faster than the alcohol formation. 

M. Temkin, Adsorption Equilibrium and the Kinetics of Processes on Non-Homogeneous Surfaces and in the Interaction Between Adsorbed Molecules, Zkumal Fizichesko i Khimii 15 296 (1941). 

A variety of models have been derived to describe the kinetics of semiconductor photocatalysis, but the most commonly used model is the Langmuir-Hinshel-wood (LH) model [77-79]. The LH model relates the rate of surface-catalyzed reactions to the surface covered by the substrate. The simplest representation of the LH model [Eq. (7)] assumes no competition with reaction by-products and is normally applied to the initial stages of photocatalysis under air- or oxygen-saturated conditions. Assuming that the surface coverage is related to initial concentration of the substrate and to the adsorption equilibrium constant, K, tire initial... 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

The next problem of the Langmuir-Hinshelwood kinetics, the validity of the rate-determining step approximation, has not been rigourously examined. However, as has been shown (e.g. refs. 57 and 63), the mathematical forms of the rate equations for the Langmuir-Hinshelwood model and for the steady-state models are very similar and sometimes indistinguishable. This makes the meaning of the constants in the denominators of the rate equations somewhat doubtful in the Langmuir—Hinshelwood model, they stand for adsorption equilibrium constants and in the steady-state models, for rate coefficients or products and quotients of several rate coefficients. 

In the interpretation of the kinetics, it was concluded that a mechanism involving adsorption equilibrium between methylcyclohexane in the gas phase and methylcyclohexane molecules adsorbed on platinum sites was not very likely. If Eq. (1) were interpreted on such a basis, then b would be an adsorption equilibrium constant. From the temperature dependence of b, one would calculate a heat of adsorption of 30 kcal./mole, which seems too high for adsorption of methylcyclohexane molecules as such. Furthermore, the small inhibition by aromatics casts doubt on such a picture, since the extent of adsorption of aromatics would be expected to be considerably greater than that of methylcyclohexane molecules at equilibrium, in view of the unsaturated nature of aromatics. 

In contrast, physical adsorption is a very rapid process, so the rate is always controlled by mass transfer resistance rather than by the intrinsic adsorption kinetics. However, under cerlaiii conditions the combination of a diffusion-controlled process with an adsorption equilibrium constant that varies according to equation 1 can give the appearance of activated adsorption. 

X. Adsorption Equilibrium and the Kinetics of Reaching it on Nonuniform Surfaces... 

K3,K5,Kf),K1, and K8 appearing in the kinetic equations are adsorption equilibrium constants and their dependence on temperature is determined by respective heat of adsorption. Since these constants are of the same order of magnitude, the heat of adsorption should be approximately the same. Thus the temperature dependence of K3, K5, K6, Kn, and K8 can be described with some common value of the heat of adsorption. 

---