Rate expression, adsorption limiting

Figure 4.8 shows some experimental uptake curves for the adsorption of CO2 in a 5A zeolite which agree very well with equation (4.39) expressing rates of adsorption limited by heat transfer. 

How useful is the rate expression derived from collision theory for describing adsorption For cases in which adsorption is not activated, i.e. E = 0, the collision frequency describes, in essence, the rate of impingement of a gas on a surface. This is an upper limit for the rate of adsorption. In general, the rate of adsorption is lower, because the molecules must, for example, interact inelastically with the sur-... 

There are a number of limiting forms of the rate expression depending on the magnitudes of the various terms in the denominator relative to unity and to each other. Any species that is weakly adsorbed will not appear in the denominator. If species R undergoes dissociation on adsorption, the term KRPR must be replaced by >JKrPr according to the discussion in Section 6.2.1.3. If an inert specie I is also capable of adsorption, the term KjPj must be added to... 

Hougen- Watson Models for Cases where Adsorption and Desorption Processes are the Rate Limiting Steps. When surface reaction processes are very rapid, the overall conversion rate may be limited by the rate at which adsorption of reactants or desorption of products takes place. Usually only one of the many species in a reaction mixture will not be in adsorptive equilibrium. This generalization will be taken as a basis for developing the expressions for overall conversion rates that apply when adsorption or desorption processes are rate limiting. In this treatment we will assume that chemical reaction equilibrium exists between various adsorbed species on the catalyst surface, even though reaction equilibrium will not prevail in the fluid phase. 

The analyses developed in this section are readily extended to reactions with different stoichiometries. Regardless of whether an adsorption or a desorption process is rate limiting, the resulting rate expressions may be written in the typical Hougen-Watson fashion represented by equation 6.3.30. A comprehensive summary of such relations has been developed by Yang... 

It has been suggested that the rate limiting step in the mechanism is the chemisorption of propionaldehyde and that the hydrogen undergoes dissociative adsorption on nickel. Determine if the rate expression predicted by a Hougen-Watson model based on these assumptions is consistent with the experimentally observed rate expression. 

The first termination step occurs homogeneously, while the second occurs by adsorption of R on the walls of the reactor. Formulate an expression for the rate of this reaction expected in a tube of diameter D with laminar flow (Shp = 8/3) and a diffusion coefficient Dr. What effective rate expressions are obtained in the limits of when... 

Based on experiments carried out with small catalyst particles under vigorous stirring, experimental data representing intrinsic kinetics were obtained. Rate expressions based on the principle of an ideal surface, rapid adsorption and desorption, but rate-limiting hydrogenation steps were derived. The competiveness... 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

If the adsorption step itself is rate-limiting, one must have available rate expressions for the adsorption and the desorption steps. The flux in (2.108) is then split into two opposing components. Using the notation of Delahay and Mohilner [201,403], there is a forward flux vj, adding to the adsorbate s surface concentration and backward flux tadsorbed substance. These obey rate equations rather analogous to those for electron transfer, the Butler-Volmer equation, in the sense that there are rate constants that are potential dependent. For the forward and backward rates, we have... 

Note Since the model is linear for the special case considered, the same equation is also satisfied by the other three variables.) The following observations may be made from Eq. (98) that expresses the dimensionless dispersion coefficient A (i) The first term describes dispersion effects due to velocity gradients when adsorption equilibrium exists at the interface. We note that this expression was first derived by Golay (1958) for capillary chromatography with a retentive layer, (ii) The second term corresponds to dispersion effects due to finite rate of adsorption (since this term vanishes if we assume that adsorption and desorption are very fast so that equilibrium exists at the interface), (iii) The effective dispersion coefficient reduces to the Taylor limit when the adsorption rate constant or the adsorption capacity is zero, (iv) As is well known (Rhee et al., 1986), the effective solute velocity is reduced by a factor (1 + y). (v) For the case of irreversible adsorption (y — oo and Da —> oo), the dispersion coefficient is equal to 11 times the Taylor value. It is also equal to the reciprocal of the asymptotic Sherwood number for mass transfer in a circular... 

Rate expressions for adsorption-desorption of NH3 over a V205-based catalyst were provided by Noskov et al. [50]. They would assume an ideal Langmuir-type catalyst surface, which is probably adequate only for limited ranges of operating conditions. 

For adsorption-limited reactions, is small and k and are large. Consequently, the ratios r /k and rjy/ko are very small (approximately zero), whereas the ratio is relatively large. The surfaee reaction rate expression is... 

An excellent illustration of the LHHW theory is catalytic cracking of n-alkanes over ZSM-5 [8]. For this reaction, the observed activation energy decreases from 140 to -50 ( ) kj/mol when the carbon number increases from 3 to 20. The decrease appeared to linearly depend on the carbon number as shown in Fig. 3.11. This dependence can be interpreted from a kinetic analysis that showed that the hydrocarbons (A) are adsorbed weakly under the experimental conditions. The initial rate expression for a rate-determining surface reaction applies (3.30), which in the limiting case of weak adsorption of A reduces to Eqn. (3.52). The activation energy is then represented by equation (3.53). 

If surface reaction is assumed to be rate limiting and irreversible (and no adsorbed inerts are involved), the overall rate expression for consumption of A becomes -rA = A aCa/(1 + KaCa + KbCb), where k is the surface reaction rate constant and Ka and A b are adsorption equilibrium constants. If the surface is only sparsely covered, i.e., KaCa + KbCb 1, this can be approximated as simply va kKACA = k CA-This illustrates how a simple power law rate expression can apply, under some circumstances, for what is actually a relatively complex mechanism. 

In this example too, similar unexpected sizes and signs of activation energies are possible. The conclusion is that in catalytic reactions there are no readily definable limits to experimental overall activation energies or frequency factors. In order to determine if an experimentally determined overall parameter is acceptable we need to know its structure in terms of adsorption and rate constants from a mechanistic formulation of the full rate expression. Only then can we hope to be able to assign estimated values to the Arrhenius parameters of these more elementary constants. We can then compare the observed values of the composite constant with those that are justified when acceptable parameter values are assigned to the constituent elementary constants. 

Limiting, Non-Zero Growth Rate Behavior. One of the most generally applicable growth rate expressions, developed for either kink, step, or ledge adsorption, is given as (Bliznakov 1959, 1965)... 

The initial BSA adsorption rate onto PDMS, PD< S and PCPMS is transport-limited for wall shear rates ranging from 25 to 4000 s. BSA adsorption on PSS is transport-limited below 70 s but becomes kinetically-limited at increasing shear rates. No adsorption (on the time scale of the experiment) occurs on PEO. Only on PMMA is BSA adsorption kinetically limited over the entire range of shear rates investigated. The initial adsorption of BSA on PMMA can be described by a kinetic rate expression that is first order in BSA concentration. 

Step 1. Reactants enter a packed catalytic tubular reactor, and they must diffuse from the bulk fluid phase to the external surface of the solid catalyst. If external mass transfer limitations provide the dominant resistance in this sequence of diffusion, adsorption, and chemical reaction, then diffusion from the bulk fluid phase to the external surface of the catalyst is the slowest step in the overall process. Since rates of interphase mass transfer are expressed as a product of a mass transfer coefficient and a concentration driving force, the apparent rate at which reactants are converted to products follows a first-order process even though the true kinetics may not be described by a first-order rate expression. Hence, diffusion acts as an intruder and falsifies the true kinetics. The chemical kineticist seeks to minimize external and internal diffusional limitations in catalytic pellets and to extract kinetic information that is not camouflaged by rates of mass transfer. The reactor design engineer must identify the rate-limiting step that governs the reactant product conversion rate. 

Rate expressions of this type have been proposed to correlate the data for the reaction between hydrogen and carbon dioxide on tungsten. Because adsorption occurs independently on each type of site and displacement of one species by another does not occur, this type of reaction will not exhibit a maximum of the type noted in case II. Instead, the rate will increase with increasing partial pressure of one reactant, eventually approaching an asymptotic limit corresponding to saturation of the type of site in question. 

In this system, glucose is postulated to form a complex with a carrier molecule at the outer surface of the cell. The sugar-carrier complex passes across the membrane and releases free glucose at the inner surface. The process is reversible. The maximum transport rate, Tmax, is limited by fixed properties of the system such as the total number of carriers and their movement. Below this limit, however, transport will vary with the sugar concentration since this determines the extent of complex formation according to Langmuir adsorption or Michaelis-Menten kinetics. Thus unidirectional transport into the cell can be expressed as follows ... 

It was early recognised that the rate limiting step in the ammonia synthesis is the dissociative adsorption of nitrogen (23) and that hydrogenation proceeds at a much faster rate (24). Temkin and Pyzhev (25) proposed a rate expression. 

Thus, in the limit of a strong surface heterogeneity leading to the Temkin equilibrium isotherm, the assumption of a strongly "activated" adsorption leads us to the rate expression... 

The power law expression was widely adopted in the literature for CO oxidation [25-27]. This form is simplified from a Langmuir-Hinshelwood (L-H) expression and not suitable for small CO concentrations [30]. Therefore a full L-H expression for CO oxidation is necessary to account for a wide range of CO concentrations (Equation 27.4). The H2 oxidation was previously modeled using empirical power law rate expressions by others [29]. However, in PrOx in the presence of CO, the rate-limiting CO desorption strongly inhibits H2 and O2 adsorption and the subsequent H2 oxidation. Hence the incorporation of Pco in the H2 oxidation rate expression is necessary (Equation 27.5). The kinetics of the r-WGS reaction were well studied previously [31], in which an empirical reversible rate expression [32] is attractive due to its relative simplicity and its appropriateness in PrOx kinetic studies, as demonstrated previously [29]. 

The parameters estimates and the confidence limits of the estimates at 95 % probability for de chosen model (model 14 in Table 2), are reported in Table 3. As can be seen, the signs of the obtained activation energies (Ea) have physicochemical significance. The equilibrium constants of adsorption K which appear in the rate expressions can be written as ... 

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