Pseudo-first-order treatment

Pseudo-First-Order Treatment. A convenient simplification is possible when the concentrations of reactants B and C are much greater than the substrate A. As the reaction progresses, changes in [B] and [C] are small relative to changes in A], and the former can be considered effectively constant. Each elementary reaction can then be considered pseudo-first-order with respect to A. Let... 

Herein k js is the observed pseudo-first-order rate constant. In the presence of micelles, analogous treatment of the experimental data will only provide an apparent second-order rate constant, which is a weighed average of the second-order rate constants in the micellar pseudophase and in the aqueous phase (Equation 5.2). 

The isolation technique showed that the reaction is first-order with respect to cin-namoylimidazole, but treatment of the pseudo-first-order rate constants revealed that the reaction is not first-order in amine, because the ratio k Jc is not constant, as shown in Table 2-2. The last column in Table 2-2 indicates that a reasonable constant is obtained by dividing by the square of the amine concentration hence the reaction is second-order in amine. For the system described in Table 2-2, we therefore find that the reaction is overall third-order, with the rate equation... 

We can reach two useful conclusions from the forms of these equations First, the plots of these integrated equations can be made with data on concentration ratios rather than absolute concentrations second, a first-order (or pseudo-first-order) rate constant can be evaluated without knowing any absolute concentration, whereas zero-order and second-order rate constants require for their evaluation knowledge of an absolute concentration at some point in the data treatment process. This second conclusion is obviously related to the units of the rate constants of the several orders. 

Reactions catalyzed by hydrogen ion or hydroxide ion, when studied at controlled pH, are often described by pseudo-first-order rate constants that include the catalyst concentration or activity. Activation energies determined from Arrhenius plots using the pseudo-first-order rate constants may include contributions other than the activation energy intrinsic to the reaction of interest. This problem was analyzed for a special case by Higuchi et al. the following treatment is drawn from a more general analysis. ... 

With two of the concentrations in large excess, the fourth-order kinetic expression has been reduced to a first-order one, with considerable mathematical simplification. The experimental design in which all the concentrations save one are set much higher, so that they can be treated as approximate constants, is termed the method of flooding (or the method of isolation, since the dependence on one reagent is thereby isolated). We shall consider the method of flooding further in Section 2.7. Here our concern is with the data analysis it should be evident that the same treatment suffices for first-order and pseudo-first-order kinetics. 

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

The quantity and quality of experimental information determined by the new techniques call for the use of comprehensive data treatment and evaluation methods. In earlier literature, quite often kinetic studies were simplified by using pseudo-first-order conditions, the steady-state approach or initial rate methods. In some cases, these simplifications were fully justified but sometimes the approximations led to distorted results. Autoxidation reactions are particularly vulnerable to this problem because of strong kinetic coupling between the individual steps and feed-back reactions. It was demonstrated in many cases, that these reactions are very sensitive to the conditions applied and their kinetic profiles and stoichiometries may be significantly altered by changing the pH, the absolute concentrations and concentration ratios of the reactants, and also by the presence of trace amounts of impurities which may act either as catalysts and/or inhibitors. 

The only reactions considered so far have been those that proceed to all intents and purposes (>95%) to completion. The treatment of revers/We reactions is analogous to that given above, although now it is even more important to establish the stoichiometry and the thermodynamic characteristics of the reaction. A number of reversible reactions are reduced to pseudo first-order opposing reactions when reactants or products or both are used in excess... 

If in the relaxation systems listed in Table 1.2 one of the reactants A or B and one of the products C or D is in large excess, that is if pseudo first-order conditions obtain, the relaxation expression is identical with the rate law obtained starting from pure reactants (1.148). For conditions other than these however, the simplified treatment with relaxation conditions is very evident, as can be seen, for example, in the simple expression for the first-order relaxation rate constant for the A -I- B C -i- D scheme compared with the treatment starting from only A and B, and when pseudo first-order conditions cannot be imposed.""... 

In batch kinetic tests, Yan and Schwartz (1999) investigated the oxidative treatment of chlorinated ethylenes in groundwater using potassium permanganate. 1,1-Dichloroethylene reacted more quickly than cis- and /ra/ 5-l, 2-dichloroethylene, trichloroethylene, and tetrachloroethylene. The reaction rate decreased with an increasing number of chlorine substituents. The pseudo-first-order rate constant and half-life for oxidative degradation (mineralization) of 1,1-dichloroethyene were 2.38 x 10 Vsec and 4.9 min, respectively. 

PhC properties most investigated by scientists to date are their water solubility (s, mg/mL), volatility (correlated to the Henry constant H) (pg m atr/pg m wastewater), biodegradability (correlated to pseudo-first-order degradation constant bioi L gSS d ), acid dissociation constant K, distribution and sorption (through the sludge-water distribution coefficient K, expressed in L gSS or the octanol-water partition coefficient Kg ). The main focus has been to find any correlations between these parameters and to determine PhC removal rates during the different treatment steps. Thus, different properties have been quantified for many compounds, and software, such as EPl Suite 4.00 [54], consenting their estimation, is available. 

If we assume that the activity coefficients of X- and H20 are independent of the X- concentration at any given ionic strength, then the usual steady state treatment leads, without further approximation, to Equation 3, a relationship between the pseudo first-order rate constant and the other kinetic parameters. 

The lower activity of most chelate complexes than that of the aquo complex of Cu was attributed to small amounts of free Cu liberated by the labile complexes The dismutase activity reported for the [Cu(I)gCu(II)g(D-penicillamine)jjCl] complex was suppressed by EDTA or by a Chelex 100 treatment Free Cu cations can, however, not explain the very high second-order rate constant of 6x 10 M " s measured at pH 7.0 with the [Cu findomethacin) ] complex by the decay of OJ at 250 nm. With the Cu-histidine complexes, moreover, it was concluded from the influence of pH on the pseudo-first order rate constant of the dismutation of HOj/Oj that [CuHiSjH] was the active species with a second-order rate constant of 3x 10 M" s between pH 2 and 7. It was not possible kinetically to distinguish between the classical and the alternative mechanism, with respectively Cu(I) or a Cu(II)—Oj" complex as intermediate... 

The application of the UV process alone showed low efficiency, which was attributed to the presence of UV-absorbing material. In addition, the photolysis of fluorene compared with the two other PAHs was significantly retarded when the fluorene was present in surface waters however, the oxidation rates of the three PAHs were essentially independent of the water type. It was concluded that the elimination of PAHs is mainly due to HO generated from hydrogen peroxide photolysis. In order to simplify the kinetic treatment, the pseudo first-order kinetics is used to determine the... 

Table 10.7 illustrates the results of Gloyna and Li (1993) for treatment of hazardous wastes by SCWO. Pressure is given in psi, where 1 bar = 14.50 psi. The following equations describe global reaction rates for the SCWO of six different substituted phenols. The rates of phenolic compound expressions were obtained at 460°C and 250 atm with [organic] = 100 pmol/L and [02] = 7 mmol/L. This rate was calculated as pseudo first order for SCWO of each phenolic compound. 

For a pulse-type NMR experiment, the assumption has a straightforward interpretation, since the pulse applied at the moment zero breaks down the dynamic history of the spin system involved. The reasoning presented here, which leads to the equation of motion in the form of equation (72), bears some resemblance to Kaplan and Fraenkel s approach to the quantum-mechanical description of continuous-wave NMR. (39) The crucial point in our treatment is the introduction of the probabilities izUa which are expressed in terms of pseudo-first-order rate constants. This makes possible a definition of the mean density matrix pf of a molecule at the moment of its creation, even for complicated multi-reaction systems. The definition of the pf matrix makes unnecessary the distinction between intra- and inter-molecular spin exchange which has so far been employed in the literature. 

Kinetic traces are now exponential and the first-order treatment yields /cv, which will exhibit a linear dependence on [B]. The true rate constant k can then be easily obtained from the relationship/c Ar, (/ B. A similar treatment is applicable to other reaction types as well. A third-order reaction, for example, can be run under pseudo-first- or pseudo-second-order conditions, depending on the precise rate law and the chemistry involved. 

The use of the pseudo-first-order dose constant assumes that the removal of solutes is exponential, which is common in waste-treatment applications [10]. For example, the concentration of OH calculated from Eq. (16) for absorbed doses between 1 and 10 kGy is 0.28 to 2.8 mM. Under these conditions the loss of the solutes (at typical solute concentrations in the micromolar range) is pseudo-first-order, with respect to absorbed dose, and can be described by the following ... 

A more fundamental difference between isotope and e-beam sources is dose rate. Whereas the high dose rates of radiation provided by e-beams are necessary for cost-effective water treatment, they also introduce complications resulting from the very high radical concentrations produced. High radical concentrations favor radical/radical recombination, resulting in a loss of reactive species. Gehringer [52] has shown a departure from pseudo-first-order kinetics in such situations, due at least in part to dose rate. 

Fig. 19.11 2,4,5-T concentration decay during the same treatments of Fig. 19.10 at 100mA. The inset panel shows its kinetic analysis assuming a pseudo-first-order reaction for 2,4,5-T...

Kinetic analysis of all concentration decays tit well with the equation related to a pseudo-first-order reaction (see the inset panel of Fig. 19.11). The last column of Table 19.2 lists the k -values thus determined for the comparative treatments of chlorophenoxy herbicides. Similar k values can be observed for all electro-Fenton and photoelectro-Fenton treatments with Pt. This discards direct photolysis of herbicides by UVA light and significant participation of reaction (19.24) to generate OH. 

An important step towards treatment of SO2 conversion to sulfate and deposition of both species that avoids absolute uncertainties of dispersion and deposition rates was taken by Lewis and Stevens, who investigated the mathematical basis of one form of hybrid receptor modeling (.16). Their model assumes that one measures concentrations of SO2 and SO4 relative to that of some species borne by particles from the plant. They assumed that (1) dispersion, deposition and transformation of the three species (SO2, SO4 and fine primary particles) are linear or pseudo first-order processes, but may have complex dependences on time (2) dispersion affects all three pollutants identically (3) dry deposition is the only type of deposition which occurs (4) deposition velocity is the same for all fine particles, but may be different for SO2 (5) secondary sulfate is produced only by homogeneous oxidation of SO2. 

The pseudo-first order rate coefficients Xjk used here were first introduced by Christiansen [3] as "reaction probabilities" Uj. Equivalent quantities are also standard in generalized treatments of chain reactions [4,5]. 

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