Identity rates

To understand why a racemic product results from the reaction of T120 wjtl 1-butene, think about the reaction mechanism. 1-Butene is first protonaled tc yield an intermediate secondary (2°) carbocation. Since the trivalent carbon i sp2-hybridized and planar, the cation has no chirality centers, has a plane o symmetry, and is achiral. As a result, it can react with H20 equally well fron either the top or the bottom. Reaction from the top leads to (S)-2-butano through transition state 1 (TS 1) in Figure 9.15, and reaction from the bottorr leads to R product through TS 2. The two transition states are mirror images. The] therefore have identical energies, form at identical rates, and are equally likeb to occur. 

ElcB)i cannot be distinguished from E2 by this means, because it has the identical rate law Rate = A [substrate] [B ]. The rate law for (E1cB)r is different Rate =/ [subs-trate][B ]/[BH], but this is often not useful because the only difference is that the rate is also dependent (inversely) on the concentration of the conjugate acid of the base, and this is usually the solvent, so that changes in its concentration cannot be measured. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

The rapid oxidations of certain of these polyfunctional compounds by alkaline permanganate were examined as a supplement to the study of acetone oxidation (p. 314). The oxidations of acetol and pyruvaldehyde show identical rate laws of the form... 

Mechanical, physical, or chemical external irritants act not only at the place of occurrence, but the excitation can be also transferred along the whole plant [3,6-21]. The speed of transfer depends on many factors, such as the intensity of the irritation, temperature, chemical treatment, or mechanical wounding it is also influenced by previous excitations. The excitation reaction travels in both directions, from the top of a stem to roots and conversely, but not always at identical rates. The transfer of excitation has a complicated character accompanied by an internal change in cells and tissues. 

Next we randomly choose either one or the other terminal units or the central unit of our selected TCH isomer and check to see if it is a V(l) unit or an E(0) unit. If a terminal V unit is chosen we check to see if the neighboring central unit is V or E. If the central unit is also V, then we determine whether this W diad is m or r. For r and m W diads the relative reactivities of the terminal V unit chlorine are 3.5 and 4.6, respectively (see Table III). If the central unit is E, then we assume the relative reactivity of the isolated terminal units in the VEE or VEV triads to be identical to the isolated central unit in the EVE triads. This assumption is supported by the identical rates of (n-Bu)3SnH reduction observed here for the 2- and 4-chlorooctanes. Finally, we select a random number between 0.0 and 1.0. If it is smaller than the relative reactivity divided by the sum of the relative reactivities of all chlorines in the VW, EW or WE, VEV, VEE or EEV, and EVE isomers of TCH and partially reduced TCH (see Table III), then we remove the terminal chlorine (1—>0) and modify the relative reactivity of the central V unit in the selected TCH isomer, because its terminal neighbor has been changed from V to E. 

The Rh(COD)(BoPhoz)/PTA/C (AHC-2) was prepared and used to promote the hydrogenation of DMIT. Data for 5,000 TON and 10,000 TON reactions are given in Table 3. In the first set of reactions, the four hydrogenations had identical rates (TOF hr"1). With the 10,000 TON reactions the first three runs had the same reaction rate while the fourth and fifth showed about a 20% decrease, probably because the catalyst was exposed to hydrogen too long after mn number three was completed. These reactions were all run using 3% toluene in methanol. The toluene was used as a co-solvent in these reactions in order to... 

Matrix acidizing treatments are more often performed, nowadays, with sensors and data acquisition systems continuously recording the surface pressure and rate histories. According to a recently proposed methodology (15), these records can be used to compute downhole rate and pressure evolutions. The bottomhole pressure history is then compared to the theoretical response of an equivalent reservoir wherein a non-reactive fluid would have been injected according to an identical rate schedule. Following this method, the difference between both theoretical and actual pressure responses originates from the evolution of the skin of the true reservoir under the influence of the acid attack. Equation 1 is then used to derive the skin decrease from this pressure difference. 

Acid-promoted aquation of the binuclear complex Cu2L of the hexaaza macrocycle L = 2,5,8,17,20,23 -hexaaza[9.9]paracyclophane, whose half-life is of the order of a second, exhibits simple one-stage first-order kinetics. This is attributed to parallel reactions at each Cu(II) center having identical rate constants (305). The kinetics of dissociation of mono- and... 

For many cases the concentration of free radical becomes practically constant since the radical is formed and consumed at identical rates. Therefore, under steady state conditions... 

In this section, however, the stereoselective catalysis of hydrolysis will be described more fully in relation to the stereoselective nature of the enzyme reaction. According to Bunton et al. (1971b), (R)- and (S)-p-nitrophenyl O-methylmandelates [35] were hydrolyzed at an identical rate in borate buffers in the presence of CTAB, while a (R/S) mixture was hydrolyzed at a rate greater than either the (/ )- or (S)-enantiomer alone. They also reported that... 

Hindman and Jacobus (1974) showed that the reaction rate was affected by the aging period of micelles and that the rates of hydrolysis of both the racemic and optically pure esters proceeded, within experimental error, at an identical rate. Moss and Sunshine (1974) also examined the hydrolysis of [35], but no significant stereoselectivity was observed. 

The following statements can be made as far as the actual surface intermediates are concerned (25) (a) hydrogenation and cyclization of 3-methylpentene isomers require a common surface intermediate (since cyclization yields are proportional to the hydrogenation rate) (b) this intermediate seems to be identical for 3-methylpentenes and 3-methylpentane (since their cyclization rates are close to each other) and (c) the surface species in question does not keep the original geometry of the starting 3-methylpentane (because cis and trans isomers cyclized with identical rates). 

Figure 6.26 Comparison of melting dynamics for a conventional melting channel and a transverse barrier melting channel for an LDPE resin at identical rates and screw speeds. The conventional channel is in red while the barrier melting section is in black...

The diffusion-controlled process [see Eqs. (5.66), (5.68), and (5.69)] can be experimentally differentiated from the process occurring in the kinetic regime [see Eq. (5.75)] only by measnring the variations of k and with 5 , and 8o. Otherwise, the identical rate laws will not permit one to distinguish between the two mechanisms. 

Table 4 shows that benzene is formed at almost identical rates from cyclohexene, hexadiene, methylcyclohexene and cyclohexadiene react under low partial pressure over Ga-HZSM-5. which suggests that over these catalysts benzene is formed from the same intermediate. In contrast over H-ZSM-5, under identical experimental conditions, the rate of benzene formation from the hydrocarbons cited was one to two orders of magnitude lower. These results prove again that gallium plays a decisive role in aromatization. Over H-ZSM-5 the major hydrocarbon formed is methylcyclopentene from cyclohexene (ring contraction)... 

The more acidic fluorene in tert-butyl alcohol solution, or in DMSO solution, reacts by a process that involves the carbanion in equilibrium with hydrocarbon. Thus, fluorene and 9,9-dideuteriofluorene oxidize at identical rates. We have established that the oxidation of the anion of fluorene can be catalyzed by a variety of electron acceptors (v), including various nitroaromatics (18). The catalyzed oxidation rates were found to follow the rates of electron transfer measured by ESR spectroscopy in the absence of oxygen. These results established the catalyzed reaction as a free radical chain process without shedding light upon the mechanism of the uncatalyzed reaction. 

Rates for this reaction may easily be measured by disappearance of azide UV absorption. Most importantly, kinetic saturation behavior is noted with sufficient amounts of the reactants cycloaddition velocity becomes independent of substrate concentration. As is familiar from enzyme catalysis, this indicates complete occupancy of all available cucurbituril by reacting species. In actuality, the rate of the catalyzed reaction under conditions of saturation was found to be the same as that for release of the product from cucurbituril. Such a stoichiometric triazole complex was independently prepared and its kinetics of dissociation were examined by the displacement technique previously outlined, giving the identical rate constant of 1.7xl0 s under the standard conditions. (It is not uncommon for product release to be rate-limiting in enzymic reactions). 

It should be emphasized that Equation 11 and Equation 12 represent identical rates as a result of the purely neutral substitutions ksp = / and cE = cEo/(l + cs/KM). Based on the format of Equation 12, it is rather tempting to treat the specificity... 

A virtually identical rate expression to that given in eqn. (20) is obtained for the alternative postulate of gaseous A reacting with adsorbed B. 

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