Adsorption, isotherms rates

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Just as the surface and apparent kinetics are related through the adsorption isotherm, the surface or true activation energy and the apparent activation energy are related through the heat of adsorption. The apparent rate constant k in these equations contains two temperature-dependent quantities, the true rate constant k and the parameter b. Thus... 

Vapor Pressures and Adsorption Isotherms. The key variables affecting the rate of destmction of soHd wastes are temperature, time, and gas—sohd contacting. The effect of temperature on hydrocarbon vaporization rates is readily understood in terms of its effect on Hquid and adsorbed hydrocarbon vapor pressures. For Hquids, the Clausius-Clapeyron equation yields... 

The ACR Process. The first step in the SCR reaction is the adsorption of the ammonia on the catalyst. SCR catalysts can adsorb considerable amounts of ammonia (45). However, the adsorption must be selective and high enough to yield reasonable cycle times for typical industrial catalyst loadings, ie, uptakes in excess of 0.1% by weight. The rate of adsorption must be comparable to the rate of reaction to ensure that suitable fronts are formed. The rate of desorption must be slow. Ideally the adsorption isotherm is rectangular. For optimum performance, the reaction must be irreversible and free of side reactions. 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

For the case of a binary system with linear adsorption isotherms, very simple formulas can be derived to evaluate the better TMB flow rates [19, 20]. For the linear case, the net fluxes constraints are reduced to only four inequalities, which are assumed to be satisfied by the same margin /3 (/3 > 1) and so ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

The dissolution of passive films, and hence the corrosion rate, is controlled by a chemical activation step. In contrast to the enhancement of the rate of dissolution by OH ions under film-free conditions, the rate of dissolution of the passive film is increased by increasing the ion concentration, and the rate of corrosion in film-forming conditions such as near-neutral solutions follows the empirical Freundlich adsorption isotherm ... 

As with the Langmuir adsorption isotherm, which in shape closely resembles Michaelis-Menten type biochemical kinetics, the two notable features of such reactions are the location parameter of the curve along the concentration axis (the value of Km or the magnitude of the coupling efficiency factor) and the maximal rate of the reaction (Vmax). In generic terms, Michaelis-Menten reactions can be written in the form... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

Assuming adsorption to behave according to the Langmuir adsorption isotherm, we get Eq. (1.22b). Both the rate constant of proton activation and the equilibrium constant of adsorption K q depend on cavity details. 

However, with an inhomogeneous electrode surface and adsorption energies that are different at different sites, the reaction rate constant and the related parameter will also assume different values for different sites. In this case the idea that the reaction rate might be proportional to surface concentration is no longer correct. It was shown by M. Temkin that when the logarithmic adsorption isotherm (10.15) is obeyed, the reaction rate will be an exponential function of the degree of surface coverage by the reactant ... 

Analytic expressions were obtained for the O and OH adsorption isotherms by solving the steady state rate equations. 

For instance, the time course of SPE demonstrates that the solvent phase surfactant concentration steadily decreases (Fig. 3) [58]. The w/o-ME solution s water content decreases at the same rate as the surfactant [58]. The protein concentration at first increases, presumably due to the occurrence of Steps 2 and 3 above, but then decreases due to the adsorption of filled w/o-MEs by the solid phase (Fig. 3) [58]. Additional evidence supporting the mechanism given above is the occurrence of a single Langmuir-type isotherm describing surfactant adsorption in the solid phase for several SPE experiments employing a given protein type (Fig. 4) [58]. Here, solid-phase protein molecules can be considered as surfactant adsorption sites. Similar adsorption isotherms occurred also for water adsorption [58]. 

Catalyst characterization - Characterization of mixed metal oxides was performed by atomic emission spectroscopy with inductively coupled plasma atomisation (ICP-AES) on a CE Instraments Sorptomatic 1990. NH3-TPD was nsed for the characterization of acid site distribntion. SZ (0.3 g) was heated up to 600°C using He (30 ml min ) to remove adsorbed components. Then, the sample was cooled at room temperatnre and satnrated for 2 h with 100 ml min of 8200 ppm NH3 in He as carrier gas. Snbseqnently, the system was flashed with He at a flowrate of 30 ml min for 2 h. The temperatnre was ramped np to 600°C at a rate of 10°C min. A TCD was used to measure the NH3 desorption profile. Textural properties were established from the N2 adsorption isotherm. Snrface area was calcnlated nsing the BET equation and the pore size was calcnlated nsing the BJH method. The resnlts given in Table 33.4 are in good agreement with varions literature data. 

Chang, C. and Lenhoff, A.M., Comparison of protein adsorption isotherms and uptake rates in preparative cation-exchange materials, ]. Chromatogr. A, 827, 281, 1998. 

Heterogeneous chemical reactions in which adsorbed species participate are not pure chemical reactions, as the surface concentrations of these substances depend on the electrode potential (see Section 4.3.3), and thus the reaction rates are also functions of the potential. Formulation of the relationship between the current density in the stationary state and the concentrations of the adsorbing species in solution is very simple for a linear adsorption isotherm. Assume that the adsorbed substance B undergoes an... 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

The Langmuir adsorption isotherm provides a simple mechanistic picture of the adsorption process and gives rise to a relatively simple mathematical expression. It can also be used to obtain a crude estimate of specific surface areas. More important, from the viewpoint of the chemical engineer, it serves as a point of departure for formulating rate expressions for heterogeneous catalytic reactions. 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

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