Step initiation rate

In cases where an acceptable kinetic mechanism can be established, it may be possible to obtain expressions, such as those in Sect. 2, which predict concentration changes with time when the values for the rate coefficients are known. However, the use of these expressions to evaluate rate coefficients from experimental data is not always straightforward, particularly with coupled reaction systems where a key reactant participants in a reversible step. Initial rate measurements are often of insufficient accuracy and, with very complex sj stems, it becomes necessary to obtain a great deal of data from experiments in which initial concentrations can be varied. 

Fig. 24 Step initiation rate i (open symbols) and step propagation rate v (solid symbols) on the 100 faces as a function of crystallization temperature for n-C246H494 crystals growing from a 4.75% octacosane solution. Key squares, extended chain triangles, folded-chain growth (from [29])...

Step Initial rate (molec/sec X cm X10" ) Activation energy (kJ/mol)... 

In any application of a copolymer the rate of formation of the product, its molecular weight, and the uniformity of its composition during manufacture are also important considerations. While the composition of a copolymer depends only on the relative rates of the various propagation steps, the rate of formation and the molecular weight depend on the initiation and termination rates as well. We shall not discuss these points in any detail, but merely indicate that the situation parallels the presentation of these items for homopolymers as given in Chap. 6. The following can be shown ... 

Autoca.ta.Iysis. The oxidation rate at the start of aging is usually low and increases with time. Radicals, produced by the homolytic decomposition of hydroperoxides and peroxides (eqs. 2—4) accumulated during the propagation and termination steps, initiate new oxidative chain reactions, thereby increasing the oxidation rate. 

The acid-catalyzed hydrolysis of 2-alkoxy-2-phenyl-l,3-dioxolanes has been studied. The initial step is rate-determining under eertain eonditions and is deseribed by the rate law given below, whieh reveals general acid catalysis. 

If the step is initially prepared to be straight, it relaxes to its fluctuating shape in the due course of time. This time evolution of step width depends on the relaxation kinetics, and can be used to determine the values of various kinetic coefficients [3,16-18,64-66], For example, if the attachment and detachment kinetics of adsorbed atoms at a step is rate limiting, the step width increases as [65]... 

Chain reactions, 181 branching, 189 initiation step, 182 propagation steps, 182 rate laws for, 188 termination step, 182 well-behaved, 187 Chemical mechanism, 9 Chemical relaxation, 255-260 Coalescence temperature, 262 Col, 170... 

Induced reactions, 102 Induction period, 72 Inhibitor competitive, 92 noncompetitive, 93 Initial rates, method of, 8, 32 from power series, 8 Initiation step, 182 Inverse dependences, 130-131 Isokinetic relationship, 164—165 Isokinetic temperature, 163, 238 Isolation, method of (see Flooding, method of)... 

More recently, Grubbs et al. obtained a refined mechanistic picture of the initiating step by conducting a 31P NMR spectroscopic study of the phosphine exchange in precatalysts 12-A. These investigations revealed that substitution of the phosphine proceeds via a dissociative-associative mechanism, i.e., a 14-electron species 12-B is involved that coordinates the alkene to give a 16-electron species 12-C (Scheme 12) [26a]. Increased initiation rates are observed if the substituents R and the phosphine ligands PR3 in precatalysts... 

TEMPERATURE FOR SEMI-BATCH AND FINISHING MONOMER FEED RATE DURING SEMI-BATCH STEP INITIATOR FEED RATE DURING SEMI-BATCH STEP TOTAL INITIAL LOADING TIME FOR SEMI-BATCH STEP MONOMER WT. % AT START OF FINISHING INITIATOR WT. PERCENT AT START OF FINISHING TIME FROM START OF... 

The initial rate of the model reaction follows a first-order dependence for the activated catalyst, the Michael donor, and the Michael acceptor. The rate determining step is not the C-C bond formation or protonolysis but the decomplexation of the bidentate product. This was evidenced by the relationship between the initial conversion and the reaction time. Extrapolation to fg = 0 h provides a positive intercept. In other words, upon addition of the reagents, the C-C bond formation occurs almost instantaneously. The amount of product at fo correlates within the experimental error to the double precatalyst loading since the dimeric precatalyst forms two active monomeric catalyst species. 

There are two distinctly different mechanisms for a surface reaction between two species [8], for example toluene (T) and an active surface species ( ). In the Langmuir-Hinshelwood (LH) mechanism, reaction occurs between toluene emd the active surface species when both are adsorbed on the catalyst surface. If this step is the slow initiation step, the rate is proportional to the product of the coverages of toluene and the active site species ... 

Phosphine dissociation is not rate determining, resulting in a high initiation rate. The rate determining step is the coordination of ihe olefin that is competing with recoordination... 

Referring to Fig. 1.4, the solution begins with the initial concentration conditions Aq, Bq, Cq and Dq, defined at time t = 0. Knowing the magnitudes of the kinetic rate constants k], k2, k3 and k4, thus enables the initial rates of change dCA/dt, dCfi/dt, dCc/dt and dCo/dt, to be determined. Extrapolating these rates over a short period of time At, from the initial conditions, Aq, Bq, Cq and Do, enables new values for A, B, C and D to be estimated at the new time, t = t -I- At. If the incremental time step At is sufficiently small, it is assumed that the error in the new estimated values of the concentration. A, B, C and D, will also be small. This procedure is then repeated for further small increments of time until the entire concentration versus time curves have been determined. 

Which reaction channel is followed depends on the availability of neighboring free places. This could explain the effect of methanol concentration on adsorbate composition. It has been observed that the initial rate of adsorption is strongly enhanced by increasing methanol concentration [14]. From the adsorption steps given above, the first one, Eq. (2.7), is directly affected by the bulk concentration. At high methanol concentrations the Pt surface becomes very quickly covered with species like CH2OH or CHOH. Further reaction to a more stable state such as COH is inhibited because of the lack of free adjacent sites. Under these conditions CO should be formed with a greater probability. 

Table 1 illustrates the effect of hydrogen pressure on the selectivity to MIBK based on the initial rate of MIBK production and the rate of IPA production. Palladium gives very high selectivity to MIBK, typically in excess of 93% with the selectivity improving significantly with decreasing pressure. This result is of particular importance since the CD process for MIBK production is carried out at relatively low pressure (< IMPa). In contrast, alternative one-step processes for MIBK production are carried out in the liquid phase in trickle-bed reactors at pressures as high as 10 MPa. 

Mathur and Thodos Chem. Eng. Sci., 21 (1191), 1966] used the initial rate approach to analyze the kinetics of the catalytic oxidation of sulfur dioxide. They summarized the most plausible rate controlling steps for the reaction as ... 

The reduction steps on active Co sites are strongly affected by activated hydrogen transferred from promoter metal particles (Pt and Ru). Several indications for the existence and importance of hetero-bimetallic centers have been obtained.63 [Cp Co(CO)2] in the presence of PEt3 and Mel catalyzes the carbonylation of methanol with initial rates up to 44 mol L 1 h 1 before decaying to a second catalytic phase with rates of 3 mol L 1 h-1.64 HOAc-AcOMe mixtures were prepared by reaction of MeOH with CO in the presence of Co(II) acetate, iodine, and additional Pt or Pd salts, e.g., [(Ph3P)2PdCl2] at 120-80 °C and 160-250 atm.65... 

Flocculation rate limitation. The adsorption step was rate limiting for the overall flocculation process in this system. Polymer adsorption rate measurements for dispersed systems reported in the literature (2,26) do not lend themselves to direct comparisons with the present work due to lack of information on shear rates, flocculation rates, and particle and polymer sizes. Gregory (12) proposed that the adsorption and coagulation halftimes, tA and t, respectively, should be good indications of whether or not the adsorption step is expected to be rate limiting. The halftimes, tA and t, are defined as the times required to halve the initial concentrations of polymer and particles, respectively. Adsorption should not limit the flocculation rate if... 

These reactions result in an additional route of chain propagation, which allows one to exceed the rate limit due to the mechanism of action of only variable-valence ions. In fact, the initial rate of RH transformation in the presence of the cobalt bromide catalyst is determined by the rate of two reactions, namely, R02 with RH (kp) and R02 with Co2+ (kp), followed by the reactions of Co3+ with Br and Br with RH. The general scheme proposed by Zakharov includes the following steps (written in the simplified form) [206] ... 

Depending on the oxidation conditions and its reactivity, the inhibitor InH and the formed radical In can participate in various reactions determining particular mechanisms of inhibited oxidation. Of the various mechanisms, one can distinguish 13 basic mechanisms, each of which is characterized by a minimal set of elementary steps and kinetic parameters [38,43 15], These mechanisms are described for the case of initiated chain oxidation when the initiation rate v = const, autoinitiation rate fc3[ROOH] -C vy and the concentration of dissolved dioxygen is sufficiently high for the efficient conversion of alkyl radicals into peroxyl radicals. The initiated oxidation of organic compounds includes the following steps (see Chapter 2). 

Reactions of phenoxyl and aminyl radicals with RH and ROOH are chain propagation steps in oxidation inhibited by phenols and amines (see Chapter 14). Both reactions become important when their rates are close to the initiation rate (see Chapter 14). Mahoney and DaRooge [57] studied the oxidation of 9,10-dihydroanthracene inhibited by different phenols. He went on to estimate the values of rate constants ratio of the reaction of ArO with RH and the reaction In + In (reactions (9) and (10), see Chapter 14) by the kinetic study. The values of kw for the reaction... 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

The reaction rate is the rate of the elementary slow step Reaction rate = ksi0VI [AlB,] But B2 is a reaction intermediate whose concentration is difficult to determine. We can determine that concentration by assuming that the fast initial step goes equally rapidly in the forward and reverse directions. MB1-MB>1 [b.H aHb] ... 

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