Individual Rate Constants

The individual rate constants in Equations 4.4 and 4.5 are not determined by the experiments described so far. The initiation rate can be evaluated from studies using no monomer. The decomposition of a typical initiator is a first-order process  

Another way of expressing the rate constant is by specifying the time it takes for the concentration of undecomposed initiator to fall to half its original value. The half-life is 

As is the case with most of the rate constants, the temperature dependence of kj can be correlated by an Arrhenius expression  

The half life can be determined by drawing a line from the given temperature through the numbered dot correspondii to the peroxide. Read the half life at the point where the line intersects the timescale. 

FIGURE 4,5 Thermal stability of organic peroxides as indicated by time for decomposition of 50% of the original charge (half-life for first-order reaction). (Data from Wallace and Tiernan Company, Lucidol Organic Peroxides, Lucidol Division, Wallace and Tiernan, Buffalo, NY, 1968.) 

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The aim of any kinetics study is to determine the individual rate constants from a reaction scheme established in conformity with the available experimental data. More specifically to the transient elongational flow problem, the kinetics calculations should be able to reproduce faithfully ... 

The individual rate constants kj j refer to the scission of a chain of i-units in length into two fragments with j and (i — j) subunits respectively. 

The normalization coefficient K is the global scission rate constant given by the contribution of all the individual rate constants for each fracture site ... 

The exact solution is in every case compared with the three approximate ones, as represented by kirnp, kss, and kpe. Table 4-2 lists the individual rate constants, the exact solution obtained from A2 and A3, and the various approximations. Figure 4-6 shows the buildup of P, with both exact and approximate solutions displayed. 

Given the following data at 12.5 °C, carry out an analysis to obtain the individual rate constants. 

The theory of radiation-induced grafting has received extensive treatment [21,131,132]. The typical steps involved in free-radical polymerization are also applicable to graft polymerization including initiation, propagation, and chain transfer [133]. However, the complicating role of diffusion prevents any simple correlation of individual rate constants to the overall reaction rates. Changes in temperamre, for example, increase the rate of monomer diffusion and monomer... 

By using the method of Levenbeig-Marquardt [4] the activation energies and frequency factors for individual rate constants are determined as given in Table 2 and the reaction orders with respect to CPD and ethylene are estimated to be 2i = 22 = 0.94, ... 

Measurements of individual rate constants for steps in the transport cycle... 

The Kmax (and K, see below) constants determined from steady-state kinetic measurements are thus seen to be complex constants containing two or more of the individual rate constants illustrated in Fig. 2. 

The above explanation of autoacceleration phenomena is supported by the manifold increase in the initial polymerization rate for methyl methacrylate which may be brought about by the addition of poly-(methyl methacrylate) or other polymers to the monomer.It finds further support in the suppression, or virtual elimination, of autoacceleration which has been observed when the molecular weight of the polymer is reduced by incorporating a chain transfer agent (see Sec. 2f), such as butyl mercaptan, with the monomer.Not only are the much shorter radical chains intrinsically more mobile, but the lower molecular weight of the polymer formed results in a viscosity at a given conversion which is lower by as much as several orders of magnitude. Both factors facilitate diffusion of the active centers and, hence, tend to eliminate the autoacceleration. Final and conclusive proof of the correctness of this explanation comes from measurements of the absolute values of individual rate constants (see p. 160), which show that the termination constant does indeed decrease a hundredfold or more in the autoacceleration phase of the polymerization, whereas kp remains constant within experimental error. 

Bamford and Dewar have adapted the latter method to the deduction of values for r, and hence to the determination of the individual rate constants kp and kt. They chose to observe the rate of polymerization by measuring the increase in viscosity with time, using for this purpose a specially designed reaction cell equipped with a viscometer. Having established by separate experiments the relation-... 

In practice, measurement of the individual rate constants or equilibrium constants for these various chemical steps requires specialized methodologies, such as transient state kinetics (see Johnson, 1992, Copeland, 2000, and Fersht, 1999, for discussion of such methods) and/or a variety of biophysical methods for measuring equilibrium binding (Copeland, 2000). These specialized methods are beyond the scope of the present text. More commonly, the overall rate of reaction progress after ES complex formation is quantified experimentally in terms of a composite rate constant given the symbol km. 

Hence, for any irreversible enzyme inactivator, we can quantify the effectiveness of inactivation using the second-order rate constant kanJKi. This constant thus becomes the key metric that the medicinal chemist can use in exploring the SAR of enzyme inactivation by a series of compounds. In terms of individual rate constants, the definitions of both nact and A) depend on the details of the mechanisms of inactivation, as will be described below. 

The effect of the substitution of a heavy-atom directly onto the nucleus of aromatic compounds (internal heavy-atom effect) on intercombinational radiative and nonradiative processes can be seen by examination of experimental data obtained for naphthalene and its derivatives. The data obtained by Ermolaev and Svitashev<104) and analyzed by Birks(24) to obtain individual rate constants for the various processes are collected in Table 5.9. 

Crosslinking of many polymers occurs through a complex combination of consecutive and parallel reactions. For those cases in which the chemistry is well understood it is possible to define the general reaction scheme and thus derive the appropriate differential equations describing the cure kinetics. Analytical solutions have been found for some of these systems of differential equations permitting accurate experimental determination of the individual rate constants. 

What is the resultant expression for the overall rate of consumption of oxygen in terms of the individual rate constants and the concentrations of stable species Experimentally, the reaction kinetics follow the expression ... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

A comparison of equations 7.3.43 and 7.3.38 shows that they are of the same mathematical form. Both can be written in terms of four measurable kinetic constants in the manner of equation 7.3.40. Only the relationship between the kinetic constants and the individual rate constants differs. Thus, no distinction can be made between the two mechanisms using steady-state rate studies. In general, the introduction of unimolecular steps involving only isomerization between unstable intermediate complexes does not change the form of the rate expression. 

The absolute values of the individual rate constants ks and kp for the nucleophile addition and proton transfer reactions. 

A mechanism provides a description of individual chemical steps that make up the overall reaction. How fast each reaction occurs is governed by the rate constant for the reaction. The observable kinetic constants Km and Vmax are related to the individual rate constants for the individual steps by a bunch of algebra. 

Suppose that the reaction between A and B to give the intermediate is very fast and very favorable. If we have more B than A to start with, all the A is converted instantly into the intermediate. If we re following P, what we observe is the formation of P from the intermediate with the rate constant k2. If we increase the amount of B, the rate of P formation won t increase as long as there is enough B around to rapidly convert all the A to the intermediate. In this situation, the velocity of P formation is independent of how much B is present. The reaction is zero-order with respect to the concentration of B. This is a special case. Not all reactions that go by this simple mechanism are zero-order in B. It depends on the relative magnitudes of the individual rate constants. At a saturating concentration of substrate, many enzyme-catalyzed reactions are zero-order in substrate concentration however, they are still first-order in enzyme concentration (see Chap. 8). 

Central to catalysis is the notion of the catalytic site. It is defined as the catalytic center involved in the reaction steps, and, in Figure 8.1, is the molybdenum atom where the reactions take place. Since all catalytic centers are the same for molecular catalysts, the elementary steps are bimolecular or unimolecular steps with the same rate laws which characterize the homogeneous reactions in Chapter 7. However, if the reaction takes place in solution, the individual rate constants may depend on the nonreactive ligands and the solution composition in addition to temperature. 

The second and third relaxation processes were coupled, where the observed rate constants differed by a factor of 3 to 7 and the rate constant for each relaxation process varied linearly with the DNA concentration.112 This dependence is consistent with the mechanism shown in Scheme 2, where 1 binds to 2 different sites in DNA and an interconversion between the sites is mediated in a bimolecular reaction with a second DNA molecule. For such coupled kinetics, the sum and the product of the two relaxation rate constants are related to the individual rate constants shown in Scheme 2. Such an analysis led to the values for the dissociation rate constants from each binding site, one of the interconversion rate constants and the association rate constant for the site with slowest binding dynamics (Table 2).112 The dissociation rate constant from one of the sites was similar to the values that were determined assuming a 1 1 binding stoichiometry (Table 1). 

The non-linear dependence of the relaxation process on the DNA concentration was also observed in stopped-flow experiments and the same mechanism, i.e. fast pre-equilibrium followed by a slow intercalation step, was proposed." This latter study did not report values for the individual rate constants. The mechanism proposed in Scheme 4 was employed in subsequent studies despite the criticism on the accuracy for the data related to the fast kinetic component (see below). The original temperature jump study also showed that the relaxation kinetics depend on the structure of the DNA.117 The slower intercalation rate for 5 with T2 Bacteriophage DNA when compared to ct-DNA was ascribed to the glucosylation of the former DNA (Table 3). 

The kinetics for all guests were fit to the sum of two exponentials. The recovered observed rate constants differed by a factor of 4 for the kinetics of these guests with ct-DNA and by a factor of 10 for the kinetics of the guests with the polydeoxynuc-leotides. For this reason, the kinetics were analyzed by determining an apparent observed rate constant defined by the fractional amplitudes (A,) and the individual rate constants ... 

To determine the activities for the various Lal+( OR) we analyze the k2bs data as a linear combination of individual rate constants (Equation 8), where ki4, kf2... " are the second-order rate constants for each La2+( OR) promoting ethanolysis and methanolysis of 1 and 2 respectively. 

The principle of the computation is to use the expressions of the forward and backward rate constant as being those of individual rate constants and sum these individual rate constants over all electronic states weighting the contribution of each state according to the Fermi-Dirac distribution.44 Assuming that H, and the density of states and therefore Kei, are independent of the energy of the electronic states,45 the results are expressed by the following equations (see Section 6.1.8) ... 

Again, using equations (3.16) and (3.17) the individual rate constants k and k2 can be evaluated. 

Many reactions which seem to be quite simple are indeed very complex. The reactions proceed in different steps. In such stepwise complex reactions, the overall reaction rate is determined by the slowest step among different steps. Various intermediate or unstable species are produced in different steps. Thus, a reaction involving many steps will lead to complex equations. In order to express the overall rate of a complex reaction in terms of the individual rate constants, a special treatment is required. In simple procedure, the intermediates such as the atoms and free radicals, the concentrations of... 

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