Exponential relationships rate constant

From the Arrhenius form of Eq. (70) it is intuitively expected that the rate constant for chain scission kc should increase exponentially with the temperature as with any thermal activation process. It is practically impossible to change the experimental temperature without affecting at the same time the medium viscosity. The measured scission rate is necessarily the result of these two combined effects to single out the role of temperature, kc must be corrected for the variation in solvent viscosity according to some known relationship, established either empirically or theoretically. 

His relationship of the rate constant k with temperature T in Kelvin involved a constant A known as the pre-exponential factor and the activation energy Ea ... 

Recently, Orosz et al. [136] reviewed and critically reevaluated some of the known mechanistic studies. Detailed mathematical expressions for rate constants were presented, and these are used to derive relationships, which can then be used as guidelines in the optimization procedure of the POCL response. A model based on the time-window concept, which assumes that only a fraction of the exponential light emission curve is captured and integrated by the detector, was presented. Existing data were used to simulate the detector response for different reagent concentrations and flow rates. 

For a two-compartment model C=0 and the equation is bi-exponential. The exponents o, jSand y are related to the intercompartmental transfer rate constants by complex formulae. They are related to the half-lives for each of the distribution and terminal phases by the relationship ... 

An attempt is often made to relate T] and T2 to the molecular dynamics of a system. For this purpose a relationship is sought between T1 or T2 and the correlation time tc of the nuclei under investigation. The correlation time is the time constant for exponential decay of the fluctuations in the medium that are responsible for relaxation of the magnetism of the nuclei. In general, l/xc can be thought of as a rate constant made up of the sum of all the rate constants for various independent processes that lead to relaxation. One of the most important of these (1 /t2) is for molecular tumbling. 

If pure A is placed in a solution, its concentration will decrease until it reaches an equilibrium with the B which has been formed. It is easy to show that in this case [A] does not decay exponentially but [A] - [A]equil does. If log([A] - [A]equil) is plotted against time a first-order rate constant k, characteristic of the rate of approach to equilibrium, will be obtained. Its relationship to fcjand k2 is given by Eq. 9-11. 

As discussed in Chapter 1 (Sections III and TV), the kinetics of drug degradation has been the topic of numerous books and articles. The Arrhenius relationship is probably the most commonly used expression for evaluating the relationship between rates of reaction and temperature for a given order of reaction (For a more thorough treatment of the Arrhenius equation and prediction of chemical stability, see Ref. 13). If the decomposition of a drug obeys the Arrhenius relationship [i.e., k = A exp(—Ea/RT), where k is the degree of rate constant, A is the pre-exponential factor ... 

Previous studies on paraffins, rhodamine dyes, and l,3-bis(N-carbozoyl) propane excimers have concluded that there is a relationship between km and polymer viscosity and free volume [103-105], Indeed, this dependence has been investigated in the context of decreasing free volume during methyl methacrylate polymerization [83,84], It has been shown that the nonradiative decay processes follow an exponential relationship with polymer free volume (vf), in which kra reduces as free volume is decreased [see Eq. (5)]. Here, k. represents the intrinsic rate of molecular nonradiative relaxation, v0 is the van der Waals volume of the probe molecule, and b is a constant that is particular to the probe species. Clearly, the experimentally observed changes in both emission intensity and lifetime for/ac-ClRe(CO)3(4,7-Ph2-phen) in the TMPTA/PMMA thin film are entirely consistent with this rationale. 

The Swedish chemist and 1903 Nobel laureate Svante August Arrhenius formalized the relationship between k and Tas the Arrhenius equation (Eq. (2.4)), where A is the pre-exponential factor (sometimes also called the frequency factor), R is the universal gas constant, roughly 2 cal mol-1 K- Tis the absolute temperature, and Ea is the reaction activation energy [5]. Although the Arrhenius equation stems from empirical observations, it explains well the temperature dependence of many rate constants over a wide range of temperatures. 

Kinetic traces are now exponential and the first-order treatment yields /cv, which will exhibit a linear dependence on [B]. The true rate constant k can then be easily obtained from the relationship/c Ar, (/ B. A similar treatment is applicable to other reaction types as well. A third-order reaction, for example, can be run under pseudo-first- or pseudo-second-order conditions, depending on the precise rate law and the chemistry involved. 

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

Although the existence of the M.I.R. may have appeared counter-intuitive to many chemists, photophysicists had a different point of view, since an inverted" relationship of the rate constant of nonradiative transitions and the energy difference between the states is well established [91]. This energy gap law results from the decreasing vibrational overlap of electronic states, the so-called Franck-Condon factor. It predicts an exponential relationship of the rate constant of nonradiative deactivation of excited states with the energy gap, of the form ... 

This relationship is strictly valid only as long as the pre-exponential factors of the involved rate constants do not vary significantly. Furthermore, in regard to Equation (12b), we find that a high chain-growth rate also requires the following relationship to be satisfied ... 

Finally, Delancey and Chiang5 3,54 reported a general mathematical evaluation of multicomponent non isothermal mass transfer in the presence as well as absence of a chemical reaction. These studies followed the matrix approach to the problem. The chemical reaction considered was a simple first-order irreversible reaction. The problem was solved assuming time dependence of the rate constant and an exponential relationship between the temperature and the distance. 

Mathematically, the combustion process has been modelled for the most general three-dimensional case. It is described by a sum of differential equations accounting for the heat and mass transfer in the reacting system under the assumption of energy and mass conservation laws At present, it is impossible to obtain an analytical solution for the three-dimensional form. Therefore, all the available condensed system combustion theories are based on simplified models with one-dimensional or, at best, two-dimensional heat and mass transfer schemes. In these models, the kinetics of the chemical processes taking place in the phases or at the interface is described by an Arrhenius equation (exponential relationship between the reaction rate constant and temperature), and a corresponding reaction order with respect to reactant concentrations. 

Equation 38E shows that the exchange current density is a function of the concentration of both reactants and products. This relationship is also implicit in Eq. 34E, considering that the rate constants in this equation depend exponentially on, which itself is a function of... 

Note than the parameters k and a are not identical with the apparent rate constant ko and the apparent charge transfer coefficient a defined by Eqs. (23) and (24), respectively. Since k can depend on the potential E indirectly through the exponential term with the potential differences across the space charge regions in Eq. (32), kl - k E = °). The relationship between a and a can be derived from Eqs. (31) and (32),... 

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