Kinetic models pseudo-first-order

It was noted from Figure 13.5 where the MB adsorption is plotted as a function of time that the free gel was saturated with the dye within 54 h and more than 70% of the dye was removed within 240 min. The sharp increase in adsorption rate, points to the fact that the adsorption takes place at the polymer surface (Hasaine et al., 2003). The adsorption rate was evaluated by two kinetic models, pseudo-first order and pseudo second order kinetic models, explained inequations 13.3 and 13.4... 

Three kinetic models, pseudo-first-order, pseudo-second-order, and intra-particle diffusion model are used to fit the experimental data The mathematical description of these models is given below. The conformity between data predicted by arty of these models and the experimental data is indicated by the correlation coefficient R. The model of higher values of R means that it successfully describes the adsorption kinetics. 

Various sets of aromatic compounds and three different molecular descriptors such as EHOmo/ Elumo, and Hammett s constants have been used for other QSAR models. First-order kinetic rates, pseudo first-order kinetic rates, and activation energy were used to correlate with different molecular descriptors. 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

For some organic compounds, such as phenols, aromatic amines, electron-rich olefins and dienes, alkyl sulfides, and eneamines, chemical oxidation is an important degradation process under environmental conditions. Most of these reactions depend on reactions with free-radicals already in solution and are usually modeled by pseudo-first-order kinetics ... 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

By lifting the simplifying restrictions, the kinetic observations can be examined in more detail over much wider concentration ranges of the reactants than those relevant to pseudo-first-order conditions. It should be added that sometimes a composite kinetic trace is more revealing with respect to the mechanism than the conventional concentration and pH dependencies of the pseudo-first-order rate constants. Simultaneous evaluation of the kinetic curves obtained with different experimental methods, and recorded under different conditions, is based on fitting the proposed kinetic models directly to the primary data. This method yields more accurate estimates for the rate constants than conventional procedures. Such an approach has been used sporadically in previous studies, but it is expected to be applied more widely and gain significance in the near future. 

Abstract Removal of catechol and resorcinol from aqueous solutions by adsorption onto high area activated carbon cloth (ACC) was investigated. Kinetics of adsorption was followed by in-situ uv-spectroscopy and the data were treated according to pseudo-first-order, pseudo-second-order and intraparticle drfiusion models. It was fotmd that the adsorption process of these compotmds onto ACC follows pseudo-second-order model. Furthermore, intraparticle drfiusion is efiective in rate of adsorption processes of these compoimds. Adsorption isotherms were derived at 25°C on the basis of batch analysis. Isotherm data were treated according to Langmuir and Freundhch models. The fits of experimental data to these equations were examined. 

Three kinetic models were applied to adsorption kinetic data in order to investigate the behavior of adsorption process of adsorbates catechol and resorcinol onto ACC. These models are the pseudo-first-order, the pseudo-second-order and the intraparticle diffusion models. Linear form of pseudo-first-order model can be formulated as... 

In the first type of study, pseudo first-order kinetics were observed in both the sediment and aqueous phases from t=0 through two half-lives in overall chlorpyrifos disappearance (total time -8 days). For these studies, computer calculations using the model illustrated in equations 7 were again used to calculate values for kj, k and kg, assuming a value of k equal to the pseudo first-order rate constant in distilled water buffered to the same pH. Values were also calculated for Obfi assuming kg 0 (equation 10) for comparison to the experimental kg values. The results of these calculations are shown in Table VII. 

Pseudo-first-order rate constants for carbonylation of [MeIr(CO)2l3]" were obtained from the exponential decay of its high frequency y(CO) band. In PhCl, the reaction rate was found to be independent of CO pressure above a threshold of ca. 3.5 bar. Variable temperature kinetic data (80-122 °C) gave activation parameters AH 152 (+6) kj mol and AS 82 (+17) J mol K The acceleration on addition of methanol is dramatic (e. g. by an estimated factor of 10 at 33 °C for 1% MeOH) and the activation parameters (AH 33 ( 2) kJ mol" and AS -197 (+8) J mol" K at 25% MeOH) are very different. Added iodide salts cause substantial inhibition and the results are interpreted in terms of the mechanism shown in Scheme 3.6 where the alcohol aids dissociation of iodide from [MeIr(CO)2l3] . This enables coordination of CO to give the tricarbonyl, [MeIr(CO)3l2] which undergoes more facile methyl migration (see below). The behavior of the model reaction closely resembles the kinetics of the catalytic carbonylation system. Similar promotion by methanol has also been observed by HP IR for carbonylation of [MeIr(CO)2Cl3] [99]. In the same study it was reported that [MeIr(CO)2Cl3]" reductively eliminates MeCl ca. 30 times slower than elimination of Mel from [MeIr(CO)2l3] (at 93-132 °C in PhCl). 

The utility of SCFs for PTC was demonstrated for several model organic reactions - the nucleophilic displacement of benzyl chloride with bromide ion (26) and cyanide ion (27), which were chosen as model reversible and irreversible Sn2 reactions. The next two reactions reported were the alkylation and cycloalkylation of phenylacetonitrile (28,29). Catalyst solubility in the SCF was very limited, yet the rate of reaction increased linearly with the amount of catalyst present. Figure 5 shows data for the cyanide displacement of benzyl bromide, and the data followed pseudo-first order, irreversible kinetics. The catalyst amounts ranged from 0.06 (solubility limit) to 10% of the limiting reactant, benzyl chloride. 

Pseudo-first-order kinetic model (Lagergren s rate equation) In this model, the kinetic rate in differential form and its analytical solution can be expressed as... 

Because of the high reactivity of hydroxyl radicals, activated complex, and chlorinated intermediates, their concentrations are extremely low at the steady state therefore, a pseudo first-order steady state can be assumed for the kinetic modeling. As a result, the steady-state concentration of the activated complex can be obtained by setting the change of its concentration to zero ... 

As discussed earlier, the effects of the meta, para, and ortho positions of chlorine on the dechlorination kinetics of monochlorophenols, dichlorophenols, and trichlorophenols during Fenton oxidation were evaluated by comparing the rate constants of the kinetic model (Tang and Huang, 1995). This study proposed a pseudo first-order steady state with respect to organic concentration. The proposed reaction pathways considered that the hydroxyl radicals would attack unoccupied sites of the aromatic ring. 

Table 10.10 summarizes the kinetic data on SCWO of organic compounds, organic mixtures, ammonia, and carbon monoxide. M and n are the constants in Equation (10.25). These kinetic data can fit the pseudo first-order reaction models proposed by Wightman (1981). The activation energy from... 

The kinetics of the hydrolysis of acetic anhydride in dilute hydrochloric acid, Scheme 1.9, may be described by a single pseudo-first-order rate constant, k, and the investigation by calorimetry combined with IR spectroscopy, as we shall see in Chapter 8, provides a clear distinction between the heat change due to mixing of the acetic anhydride into the aqueous solution and that due to the subsequent hydrolysis. This model of the reaction is sufficient for devising a safe and efficient large-scale process. We know from other evidence, of course, that the reaction at the molecular level is not a single-step process - it involves tetrahedral intermediates - but this does not detract from the validity or usefulness of the model for technical purposes. 

If the assumptions made above are not valid, and/or information about the rate constants of the investigated reactions is required, model-based approaches have to be used. Most of the model-based measurements of the calorimetric signal are based on the assumption that the reaction occurs in one single step of nth order with only one rate-limiting component concentration in the simplest case, this would be pseudo-first-order kinetics with all components except one in excess. The reaction must be carried out in batch mode (Vr = constant) in order to simplify the determination, and the general reaction model can, therefore, be written as Equation 8.14 with component A being rate limiting ... 

As pointed out in Section 8.2, most physical and chemical processes, not just the chemical transformation of reactants into products, are accompanied by heat effects. Thus, if calorimetry is used as an analytical tool and such additional processes take place before, during, or after a chemical reaction, it is necessary to separate their effects from that of the chemical reaction in the measured heat-flow signals. In the following, we illustrate the basic principles involved in applying calorimetry combined with IR-ATR spectroscopy to the determination of kinetic and thermodynamic parameters of chemical reactions. We shall show how the combination of the two techniques provides extra information that helps in identifying processes additional to the chemical reaction which is the primary focus of the investigation. The hydrolysis of acetic anhydride is shown in Scheme 8.1, and the postulated pseudo-first-order kinetic model for the reaction carried out in 0.1 M aqueous hydrochloric acid is shown in Equation 8.22 ... 

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