Rate constant dependence

We have tacitly assumed that the rate constants depend only on the last unit of the chain. In such a situation, the copolymerization is called a Markov copolymerization of first order. The special case (i), r r- = 1, is a Markov copolymerization of order zero. If reactivity also depends on the penultimate unit of the chain, the polymerization is a Markov copolymerization of second order. 

In determining the values of Ka use is made of the pronounced shift of the UV-vis absorption spectrum of 2.4 upon coordination to the catalytically active ions as is illustrated in Figure 2.4 ". The occurrence of an isosbestic point can be regarded as an indication that there are only two species in solution that contribute to the absorption spectrum free and coordinated dienophile. The exact method of determination of the equilibrium constants is described extensively in reference 75 and is summarised in the experimental section. Since equilibrium constants and rate constants depend on the ionic strength, from this point onward, all measurements have been performed at constant ionic strength of 2.00 M usir potassium nitrate as background electrolyte . 

Data for zeroth-order nitration in these various solvents are given in table 3.1. Fig. 3.1 shows how zeroth-order rate constants depend on the concentration of nitric acid, and table 3.2 shows how the kinetic forms of nitration in organic solvents depend on the reactivities of the compounds being nitrated. 

The situation changes when moving on to low temperature. Friction affects not only the prefactor but also the instanton action itself, and the rate constant depends strongly on rj. In what follows we restrict ourselves to the action alone, and for the calculation of the prefactor we refer the reader to the original papers cited. For the cusp-shaped harmonic potential... 

The equations for the Zf ro order reaction ore true for any stoichiometry, hut the relationship between the rate constants depends on the stoichhnetry. 

The units of the rate constant depend upon the overall reaction order. 

Generally, in an equation of a chemical reaction rate, the rate constant often does not change with temperature. There are many biochemical reactions that may be influenced by temperature and the rate constant depends on temperature as well. The effect of temperature on... 

The data can be analyzed by graphical or numerical fitting to Eq. (2-19). A plot of In ([B]r/tA],) versus time affords a straight line of slope kAo- The value of the rate constant depends inversely on the difference Ao. The closer the initial concentrations are to each other, the less accurately one knows A0. 

The isomerization of A to B yielded kinetic data that conformed to a first-order rate law. but the apparent first-order rate constant depended on the initial concentration of A. The authors propose competing unimolecular and bimolecular processes, and they show that the system reduces to a first-order expression when the equilibrium constant K is unity that is,... 

A applies, since none of its three component rate constants depends on /x. Scheme B, on the other hand, is expected to show a salt effect, since both k b and k4h are accelerated by electrolytes. The equations for B are... 

FIGURE 13.24 The dependence of the rate constant on temperature for two reactions with different activation energies. The higher the activation energy, the more strongly the rate constant depends on temperature. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

MPC dynamics follows the motions of all of the reacting species and their interactions with the catalytic spheres therefore collective effects are naturally incorporated in the dynamics. The results of MPC dynamics simulations of the volume fraction dependence of the rate constant are shown in Fig. 19 [17]. The MPC simulation results confirm the existence of a 4> 2 dependence on the volume fraction for small volume fractions. For larger volume fractions the results deviate from the predictions of Eq. (92) and the rate constant depends strongly on the volume fraction. An expression for rate constant that includes higher-order corrections has been derived [95], The dashed line in Fig. 19 is the value of /. / ( < )j given by this higher-order approximation and this formula describes the departure from the cf)1/2 behavior that is seen in Fig. 19. The deviation from the <[)11/2 form occurs at smaller values than indicated by the simulation results and is not quantitatively accurate. The MPC results are difficult to obtain by other means. 

Since the transition state formulation of a reaction rate expression treats the activated complex as being in equilibrium with the reactants, the resultant expression for the reaction rate constant depends similarly on the free energy difference between reactants and the activated complex. In this case equation 4.3.34 can be rewritten as... 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

Fig. 15 Plot showing how the first order rate constant depends on AGj when Y from (33) is included.

The next difficulty in comparing the predictions of Eq. (1) with experiment is that experimental values are reported in terms of either second-order rate constants for the gas-phase experiments or pseudo-first-order rate constants for the solution experiments. According to Eq. (1), neither pure reaction order is correct nor should the apparent rate constant depend on the concentration or... 

We also assume that the various rate constants depend on T in accordance with the Arrhenius equation. Thus, from equations 3.1-8 and 4.1-3,... 

In general acid-base catalysis, the observed rate constant depends on the concentrations of all acids and bases present. That is, in aqueous systems,... 

As a general statement, the reaction rate in each direction follows second-order kinetics for all the rhenium compounds studied. Moreover, the rate constants depend on the identities of L and Ly. Both findings argue for an associative (displacement) mechanism, which is also supported by the large and negative values of AS, that often reach —120 J K-1 mol-1 (39). 

Here, i is the faradaic current, n is the number of electrons transferred per molecule, F is the Faraday constant, A is the electrode surface area, k is the rate constant, and Cr is the bulk concentration of the reactant in units of mol cm-3. In general, the rate constant depends on the applied potential, and an important parameter is ke, the standard rate constant (more typically designated as k°), which is the forward rate constant when the applied potential equals the formal potential. Since there is zero driving force at the formal potential, the standard rate constant is analogous to the self-exchange rate constant of a homogeneous electron-transfer reaction. 

Free energy is the key quantity that is required to determine the rate of a chemical reaction. Within the Conventional Transition State Theory, the rate constant depends on the free energy barrier imposed by the conventional transition state. On the other hand, in the frame of the Variational Transition State Theory, the free energy is the magnitude that allows the location of the variational transition state. Then, it is clear that the evaluation of the free energy is a cornerstone (and an important challenge) in the simulation of the chemical reactions in solution... 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

The direct application of this expression in order to estimate the degree of thermomechanic destruction in connection with polymer processing is hindered because the process rate constant depends on the temperature and intensity of thermo-mechanical impact on a material. Consequently, of significant interest is the issue of selecting an attribute for characterizing the degree of destruction. Most researchers consider it worthwhile to simply use viscosity variable (r a) or characteristic viscosity variable. 

The units of k, the rate constant, depend on the overall order of a reaction. For what overall reaction order does the rate constant have the same units as the reaction rate ... 

The gel layer thickness increases linearly with the square root of dipping time indicating that indeed a slip-casting process is operative. The rate constant depends on gel structure and pore size of the support. If the modal pore size of the support is increased from 0.12 /im (type 1 support) to 0.34 /xm (type 2 support) the casting rate is decreased in accordance with theory. Typical casting rates for type 1 and type 2 supports are 4.4 /xm/s and 2.8 /xm/s, respectively for HNO3-stabilized sols with a concentration of 1.22 mol boehmite/L. 

Notice that the outlet flow rate, q, may differ from the feed rate, qf, due to density variations during the polymerization. Moreover, the rate constants depend on reactor temperature and are described by the following Arrhenius expressions. 

The cation-radicals ArH+ were detected, but they originated from the fast reaction of a one-electron transfer, which does not affect kinetic constants of the oxidation. The rate constant depends linearly on Brown s a constants of substituents (Dessau et al. 1970). All these data are in agreement with the formation of the strong polar dication of an aromatic hydrocarbon as an intermediate. Because PF salts (in particular the diacetate) are not reductants, the two-electron transfer reaction proceeds irreversibly. 

The kinetics of the oxidation of alkenes by trans- Ku [Ru (0)2(CRMe3)] +, tranx-[Ru (0)2(pytn)] +, and traw5-[Ru (0)2(N202)] + with °(Ru ranging from 0.23 V to 0.70 V vs. SCE have been investigated in acetonitrile. The rate constants depend on the redox potential of the ruthenium oxidant as well as on the oxidation potentials and... 

Specific diffusion control The rate constants depend on the size and topology of the molecule the group is bound to i.e., they depend on the translation diffusion coefficient of the species. 

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