Pseudo-first-order rate coefficient

Herein [5.2]i is the total number of moles of 5.2 present in the reaction mixture, divided by the total reaction volume V is the observed pseudo-first-order rate constant Vmrji,s is an estimate of the molar volume of micellised surfactant S 1 and k , are the second-order rate constants in the aqueous phase and in the micellar pseudophase, respectively (see Figure 5.2) V is the volume of the aqueous phase and Psj is the partition coefficient of 5.2 over the micellar pseudophase and water, expressed as a ratio of concentrations. From the dependence of [5.2]j/lq,fe on the concentration of surfactant, Pj... 

First-order and pseudo-first-order reactions are represented by the upper curve in Fig. 14-14. We note that for first-order reactions when the Hatta number is larger than about 3, the rate coefficient k can be computed by the formula... 

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

The concentration of monomers in the aqueous phase is usually very low. This means that there is a greater chance that the initiator-derived radicals (I ) will undergo side reactions. Processes such as radical-radical reaction involving the initiator-derived and oligomeric species, primary radical termination, and transfer to initiator can be much more significant than in bulk, solution, or suspension polymerization and initiator efficiencies in emulsion polymerization are often very low. Initiation kinetics in emulsion polymerization are defined in terms of the entry coefficient (p) - a pseudo-first order rate coefficient for particle entry. 

This difference is exemplified by the variation of the pseudo first order rate coefficients (Rate = k [Substrate]) for the deamination and denitrosation of N-n-butyl -nitroso acetamide at 25 C with [HpSO ] shown in Figure 1 Denitrosation becomes the domin- ant pathway above 5M H SO. (9) but the crossover acidity i lower (ca.0 5M. H SO ) ror N-nitroso-2-pyrrolidone (10) ... 

Abbreviations k = pseudo-first order rate constant R2 = correlation coefficient r° = initial reaction rate in mmole/gcataiystxh r0Pi and r°Re = reaction rate related to the amount of Pt and Re, respectively C = conversion of AcOBu at t=24 h selectivity to BuOH was above 90 % a2 g catalyst was used. 

The rate of photolytic transformations in aquatic systems also depends on the intensity and spectral distribution of light in the medium (24). Light intensity decreases exponentially with depth. This fact, known as the Beer-Lambert law, can be stated mathematically as d(Eo)/dZ = -K(Eo), where Eo = photon scalar irradiance (photons/cm2/sec), Z = depth (m), and K = diffuse attenuation coefficient for irradiance (/m). The product of light intensity, chemical absorptivity, and reaction quantum yield, when integrated across the solar spectrum, yields a pseudo-first-order photochemical transformation rate constant. 

The ions or cluster ions are thermalized by collisions with an inert carrier gas (usually helium), although often argon or even nitrogen is employed. Neutral reactant gas is added through a reactant gas inlet at an appropriate location downstream in the flow tube, and allowed to react with the injected ions. Rate coefficients, k, are determined by establishing pseudo-first-order reaction conditions in which the reactant ion concentration is small compared to the reactant neutral concentration. Bimolecular rate coefficients, k, are obtained from the slope of the natural logarithm of the measured signal intensity, /, of the reactant ion versus the flow rate (2b of reactant gas 45,48-50... 

Kinetic theory indicates that equation (32) should apply to this mechanism. Since the extent of protonation as well as the rate constant will vary with the acidity, the sum of protonated and unprotonated substrate concentrations, (Cs + Csh+), must be used. The observed reaction rate will be pseudo-first-order, rate constant k, since the acid medium is in vast excess compared to the substrate. The medium-independent rate constant is k(), and the activity coefficient of the transition state, /, has to be included to allow equation of concentrations and activities.145 We can use the antilogarithmic definition of h0 in equation (33) and the definition of Ksh+ in equation (34) ... 

A reagent in solution can enhance a mass transfer coefficient in comparison with that of purely physical absorption. The data of Tables 8.1 and 8.2 have been cited. One of the simpler cases that can be analyzed mathematically is that of a pseudo-first order reaction that goes to completion in a liquid film, problem P8.02.01. It appears that the enhancement depends on the specific rate of reaction, the diffusivity, the concentration of the reagent and physical mass transfer coefficient (MTC). These quantities occur in a group called the Hatta number,... 

Bertole et al.u reported experiments on an unsupported Re-promoted cobalt catalyst. The experiments were done in a SSITKA setup, at 210 °C and pressures in the range 3-16.5 bar, using a 4 mm i.d. fixed bed reactor. The partial pressures of H2, CO and H20 in the feed were varied, and the deactivation, effect on activity, selectivity and intrinsic activity (SSITKA) were studied. The direct observation of the kinetic effect of the water on the activity was difficult due to deactivation. However, the authors discuss kinetic effects of water after correcting for deactivation. The results are summarized in Table 1, the table showing the ratio between the results obtained with added water in the feed divided by the same result in a dry experiment. The column headings refer to the actual experiments compared. It is evident that adding water leads to an increase in the overall rate constant kco. The authors also report the intrinsic pseudo first order rate-coefficient kc, where the overall rate of CO conversion rco = kc 6C and 0C is the coverage of active... 

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. 

PRZM was applied to a hypothetical situation of a pesticide In a Georgia agricultural environment. An overall, pseudo-first-order degradation rate coefficient of 0.001 day was used, along with a series of values. A cover crop of peanuts was assumed. The simulation was done for a 900 g/ha application to a class A soil (well drained) and a class D soil (poorly drained). Movement through the root zone was simulated using rainfall records. In the hypothetical 1-ha plot, 800 g and 550 g of the pesticide leached past 60 cm In the class A and D soils, respectively, when a Kj value of 0.06 was used 40 g and 5 g leached past 60 cm In the class A and D soils, respectively, when a Kj value of 1.5 was used. These computational results support the conclusion on Kj values stated at the end of this paper. 

Data from tests at 250,275,300, and 325 C were used to calculate pseudo-first order rate constants for the formation of H2S. These data are expressed on a standard Arriienius plot (Fig. 2) for which the linear least squares coefficient of determination, r, is 0.98. The apparent activation energy calculated from the slope is 28.5 kcal/mol. This result is in excellent agreement with the recent work of Abotsi, who studied the performance of carbon-supported hydrodesulfurization catalysts (10). Using Ambersorb XE-348 carbon lo ed with sulfided ammonium molybdate (3% Mo loading) prepared by the same procedure reported here, Abotsi hydrotreated a coal-derived recycle solvent The apparent activation energy for... 

These reactions do not satisfy total mass conservahon because the mole of water is omitted as a reactant. We have also redefined a new rate coefficient as k = [H20] by grouping the nearly constant [H2O] with k. After grouping the concentrahon of the solvent [H2O] into the rate coefficient, we say that we have a pseudo-first-order rate expression. 

A kinetic analyses of the data was performed by noting the pseudo-first order loss of substrate together with selectivity. This enabled a pseudo-first order kinetic description of the two pathways to be obtained. Table 1 lists the lifetimes of 2-butanone and 2-butanol production for the various experiments. Here the lifetimes refers to the inverse of the pseudo-first order reaction rate coefficients. 

Numerous experimental combinations of process conditions (SS or US), hydrogenation gas (H2 or D2), and solvent (H2O or D2O) have been explored. A summary of combinations we have chosen for study is presented in Table 2. In this table it is seen that the experiments are labeled B1-B7 for 3B20L and P1-P6 for 14PD30L. The second column lists the experimental conditions, whereas the third column lists the initial system concentration based on 100 mM of substrate and the amount of catalyst used. The penultimate column lists the final (extent of reaction > 95%) selectivity to ketone (2-butanone or 3-pentanone) and the final column lists the pseudo-first order substrate loss rate coefficient. The dataset contained in Table 2 enables numerous conclusions to be made regarding the reaction systems. The differences in initial concentrations (e.g., 67 versus 100 M/g-cat.) arise from the chosen convenience of having similar activities and therefore comparable reaction times. 

Many of the reactions discussed in the preceding pages are in fact bimolecular processes, which would normally follow second-order kinetics. However, as aheady discussed, under the regime of LFP they behave as pseudo-first-order reactions. The corresponding rate constants and lifetimes are independent of the initial concentration of transient, and therefore knowledge of extinction coefficients and quantum yields is not needed. Further, it is not important to have a homogenous transient concentration. 

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 purpose of this appendix, in giving detail of the derivation of eqns (1.22)—(1.24), is to demonstrate a method of analysis which will be of particular use in later chapters when we discuss the local stability of a stationary-state solution. We will see here concentrations of different species evolving as the sum of a series of exponential terms which involve first-order rate constants. Later we will see similar sums of exponential terms, where the exponents, although more complicated can also be interpreted as pseudo-first-order rate coefficients. 

An ingenious application of reactions of hydrated electrons with divalent cations was developed by Jonah et al. [126]. They increased the cation concentration in the presence of a polyvinyl sulphate. At low concentrations of cation, the rate of reaction of hydrated electrons was low, but above a well-defined concentration, the pseudo-first-order rate coefficient increased linearly and rapidly with concentration. Providing it is reasonable to assume that divalent cations bound to the polymer display much lower reactivitiy with hydrated electrons, it is possible to deduce... 

The term (t) times the second-order rate coefficient, k, is the pseudo-first-order rate coefficient for reaction. In the equation above, v(t) occurs on both sides, so taking Laplace transforms shows that... 

The pseudo-first order rate constants (resp. coefficients) for the direct reaction of some compounds may almost be in the order of typical hydroxyl rate constants (kR > 10 M s ), due to high concentrations of the pollutants as well as mass transfer enhancement. For example, Sotelo et al. (1991) measured values of 6.35 106 and 2.88 106 M l s"1 for the dissociating hydroxylated phenols, resorchinol (1,3-dihydroxybenzene) and phlorogluci-nol (1,3,5-tn hydroxybenzene) respectively (pH = 8.5 and T= 20 °C). 

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