Desorption rates, computational

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

The required distribution of initial populations ntu can be obtained in the following manner (32). Let us consider a system with Ed mi = 20 kcal/ mole and Ed max = 45 kcal/mole. Assuming that kd = 1013 sec-1 and x = 1, we can calculate theoretical desorption rates dnai/dt for Ed = 20, 21, 22,..., 45 kcal/mole as a function of nBOi. With increasing temperature, 25 values of dnjdt are measured at temperatures corresponding to Ed of 20, 21, 22,. . ., 45 kcal/mole. Since the total desorption rate at any moment must be equal to the sum of the individual desorption processes, we obtain 25 linear equations. Their solution permits the computation of the initial populations of the surface sites in the energy spectrum considered, i.e. the function n,oi(Edi). From the form of this function, desorption processes can be determined which exhibit a substantial effect on the experimental desorption curve. 

Diazinon and Ronnel. The conclusion that neutral hydrolysis of sorbed chlorpyrifos is characterized by a first-order rate constant similar to the aqueous phase value is strengthened and made more general by the results for diazinon, 0,0-diethyl 0-(2-iso-propyl-4-methyl-6-pyrimidyl) phosphorothioate, and Ronnel, 0,0-dimethyl 0-(2,4,5-trichlorophenyl) phosphorothioate (10). The results for the pH independent hydrolysis at 35°C for these compounds in an EPA-26 sediment/water system (p=0.040) are summarized in Table IV. Because the aqueous (distilled) values of k for diazinon and Ronnel are similar in magnitude to the value for chlorpyrifos, and because these values were shown by the chlorpyrifos study to be slow compared to sorption/desorption kinetics, computer calculations of were not deemed necessary and were not made for these data. 

The stage of adsorption is the simplest elementary process among the other surface processes. It can be both a main process in adsorption and one of the stages of complex interface process. At least one of the adsorption or desorption stage is always presented in any surface process. In the theory of desorption process, the AC was introduced independently for the mono-and bimolecular desorption processes by different authors [107,108] in 1974. In both papers the quasi-chemical approximation has been used. Flowever, actual computations [107] have been performed at e — 0 (the collision model). They have shown that TDS slitting is caused even by a slight repulsion e <0.05 des. The expressions obtained for the desorption rates have been applied to TDS computations for H2/W(100), CO/W(210), and N2/W(100) [109,110]. 

In these hybrid simulations, coupling happened through the boundary condition. In particular, the fluid phase provided the concentration to the KMC method to update the adsorption transition probability, and the KMC model computed spatially averaged adsorption and desorption rates, which were supplied to the boundary condition of the continuum model, as depicted in Fig. 7. The models were solved fully coupled. Note that since surface processes relax much faster than gas-phase ones, the QSS assumption is typically fulfilled for the microscopic processes one could solve for the surface evolution using the KMC method alone, i.e., in an uncoupled manner, for a combination of fluid-phase continuum model parameter values to develop a reduced model (see solution strategies on the left of Fig. 4). Note again that the QSS approach does not hold at very short (induction) times where the microscopic model evolves considerably. 

A general computational scheme using orthogonal collocation on finite elements has been developed for calculation of rates of mass transfer accompanied by single or multistep reactions. The method can be used to predict enhancement in absorption or desorption rates for a wide class of industrially important situations. 

A general computational analysis which is applicable to any reaction scheme is useful as it permits direct calculation of the absorption or desorption rates once the kinetics of the various reactions and the physico-chemical parameters are specified. Such a computational package is also useful for developing a general program for design of industrial absorbers or desorbers. Computational aspects of multistep instantaneous reactions have been analyzed by Bhattacharya and Ramachandran (7 ). A general formulation and computational procedure for reactions with finite rates does not appear to have been presented and this paper is directed towards this objective. 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

As alternative expression to be used for the terms R, A, D and X, (Guo, 1996) provides a detailed description of different source and sink sub-models in the User s Guide of his Z-30 lAQ simulator. This very useful compilation of existing widely used source and sink sub-models includes 25 source and 6 sink sub-models. The 25 source sub-models include 3 for constant source, 7 for a first-order (exponential) decay source, 3 for a higher order decaying source, 1 for an instant source, 1 for a time-varying source, 2 for ambient air as an indoor source and 8 mass transfer sub-models. The 6 sink sub-models include one irreversible and five reversible sub-models. Among the five reversible sink sub-mod-els two are equilibrium models and the remaining three compute simultaneously both adsorption and desorption rates. 

The oscillations observed with but-l-ene at 150°C are made up of two different portions. One is of sinusoidal type and the other consists of relaxation jumps between low and high conversion states. The parameters, k and A 4, adsorption and desorption rate constants of but-l-ene on the platinum surface, ks the surface reaction rate constant between O2 and but-l-ene, Zq the capacity factor, change the characteristics of these oscillations. Operating conditions like reactor temperature, flow rate, volume, mass of catalyst and concentration of but-l-ene in the feed also affect these oscillations both experimentally and in computer simulations. 

Figure 6 shows the pressure dependence of diffusion coefficients calculated from permeation, sorption, and desorption rate curves for CO2 in PI2080. The average values of diffusion coefficients from sorption and desorption rate curves D y are in fair agreement with that from permeation rate curve D. The solid line in Figure 6 was computed from Equation 14... 

The dynamic motion due to rapid energy exchange for the desorption of Xe atoms from a Pd(lOO) surface will be illustrated. Figure 5.9a shows the rate of Xe desorption as predicted according to transition-state theory. Figure 5.9b compares computed molecular-dynamics rates and the transition-state rates. The open data points are the computed desorption rates for Xe atoms that are allowed to readsorb once they have passed the transition-state barrier. The filled data points ignore the possibility of readsorption. The open data points, computed from the more exact theory, always remain lower than the transition-state result. Transition-state theory and molecular dynamics predict very similar rate constants for the desorption of xenon from palladium. 

Chern [42] developed a mechanistic model based on diffusion-controlled reaction mechanisms to predict the kinetics of the semibatch emulsion polymerization of styrene. Reasonable agreement between the model predictions and experimental data available in the literature was achieved. Computer simulation results showed that the polymerization system approaches Smith-Ewart Case 2 kinetics (n = 0.5) when the concentration of monomer in the latex particles is close to the saturation value. By contrast, the polymerization system under the monomer-starved condition is characterized by the diffusion-con-trolled reaction mechanisms (n > 0.5). The author also developed a model to predict the effect of desorption of free radicals out of the latex particles on the kinetics of the semibatch emulsion polymerization of methyl acrylate [43]. The validity of the kinetic model was confirmed by the experimental data for a wide range of monomer feed rates. The desorption rate constant for methyl acrylate at 50°C was determined to be 4 x 10 cm s ... 

Stockmayer also presented solutions for the case that takes into account desorption of radicals. This solution, however, is wrong for the most important range in desorption rates. But Sto ckmayer s solution(s) lead the way for the possibility of exact mathematical solution of emulsion polymerisation kinetics at a time when digital computers were not yet very important in chemical computations. The general solution when desorption is taken into account was presented by O Toole (1965). He applied a modified form of the Smith-Ewart recursion equation that gave the solution... 

Thermal desorption spectroscopy and temperature programmed reaction experiments have provided significant insight into the chemistry of a wide variety of reactions on well characterized surfaces. In such experiments, characterized, adsorbate covered, surfaces are heated at rates of 10-100 K/sec and molecular species which desorb are monitored by mass spectrometry. Typically, several masses are monitored in each experiment by computer multiplexing techniques. Often, in such experiments, the species desorbed are the result of a surface reaction during the temperature ramp. 

The activation free energy of desorption may be computed from the rate of desorption as determined experimentally from the change in the surface potential with time. The theory of absolute rates has been applied to desorption by Eley (120) and Higuchi et cd. (107) to obtain energies and entropies of activation as a function of coverage. The rate of desorption is given by,... 

In the model of McCall and Agin (1985), picloram desorption occurs from two different soil sites. Picloram could be released rapidly from one site (Bl) characterized by rate constant k and slowly and reversibly from the other site by rate constants k2 and k-2. The fast steps may be reversible however, in the experiments of McCall and Agin (1985) the backward rate coefficient was small since this step could be described with one rate coefficient. The plotted line in Fig. 9.4 is computed from the previous model. The model also fits the data well at early desorption times (Fig, 9.5). McCall and Agin (1985) found that k values were approximately two orders of magnitude higher than k2 and k-2. Other parameters... 

Proportionality of and t Is often (but not always) an indication of a diffusion-controlled process, but such a proportionality does not have to extend over the entire time domain considered. It may happen that diffusion control is realized but that the computed D, is lower than the corresponding value in the gas phase. One possible explanation for this may be that the supply is followed by a slower surface diffusion process, which Is rate-determining. Surface diffusion coefficients D° tend to be lower than the corresponding bulk values. Such diffusion has been briefly discussed In sec. I.6.5g, under (1). When surface diffusion Is zero, the adsorbate is localized. In that case equilibration between covered and empty parts of the surface can only take place by desorption and readsorption. For D° 0 the adsorbate is mobile it then resembles a two-dimensional gas and we have already given the partition functions for one adsorbed mobile atom in sec. I.3.5d. In sec. 1.5d we shall briefly discuss the transition between localized and mobile adsorption. 

Thus, the reaction on Pd/Fsc is rate limited in the last step, the conversion of (C4H4)(C2H2) into CeHe- This is different from the Pd(lll) surface where the rate determining step for the reaction is benzene desorption. The calculations are consistent with the experimental data. In fact, on Pdi/Fsc, the computed barrier of 0.98 eV corresponds to a desorption temperature of about 300K, as experimentally observed (Fig. 1.100). On Pd (111) surfaces, the bonding of benzene is estimated to be 1.9eV. This binding is consistent with a desorption temperature of 500 K as observed for a low coverage of CeHe on Pd(lll) [489],... 

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