Exponential model kinetic data

Kinetic data analyzed using a two-phase exponential model. The data only show the results of the analysis of the fast component. The slower component was typically two orders of magnitude or more slower than the fast component (cf. Fig. 3). We have attributed the slow component to receptor misfolding (see text for details). Dissociation produced by GTPyS was always faster and to a greater extent than GDP. 

With some further assumptions, it is possible to use single frequency FLIM data to fit a two-component model, and calculate the relative concentration of each species, in each pixel [16], To simplify the analysis, we will assume that in each pixel of the sample we have a mixture of two components with single exponential decay kinetics. We assume that the unknown fluorescence lifetimes, iq and r2, are invariant in the sample. In each pixel, the relative concentrations of species may be different and are unknown. We first seek to estimate the two spatially invariant lifetimes, iq and t2. We make a transformation of the estimated phase-shifts and demodulations as follows ... 

Table 2.9 summarizes the kinetic data which were employed by Ravindranath and co-workers in PET process models. The activation energies for the different reactions have not been changed in a decade. In contrast, the pre-exponential factors of the Arrhenius equations seem to have been fitted to experimental observations according to the different modelled process conditions and reactor designs. It is only in one paper, dealing with a process model for the continuous esterification [92], that the kinetic data published by Reimschuessel and co-workers [19-21] have been used. 

Pseudo-first-order rate constants for carbonylation of [MeIr(CO)2l3]" were obtained from the exponential decay of its high frequency y(CO) band. In PhCl, the reaction rate was found to be independent of CO pressure above a threshold of ca. 3.5 bar. Variable temperature kinetic data (80-122 °C) gave activation parameters AH 152 (+6) kj mol and AS 82 (+17) J mol K The acceleration on addition of methanol is dramatic (e. g. by an estimated factor of 10 at 33 °C for 1% MeOH) and the activation parameters (AH 33 ( 2) kJ mol" and AS -197 (+8) J mol" K at 25% MeOH) are very different. Added iodide salts cause substantial inhibition and the results are interpreted in terms of the mechanism shown in Scheme 3.6 where the alcohol aids dissociation of iodide from [MeIr(CO)2l3] . This enables coordination of CO to give the tricarbonyl, [MeIr(CO)3l2] which undergoes more facile methyl migration (see below). The behavior of the model reaction closely resembles the kinetics of the catalytic carbonylation system. Similar promotion by methanol has also been observed by HP IR for carbonylation of [MeIr(CO)2Cl3] [99]. In the same study it was reported that [MeIr(CO)2Cl3]" reductively eliminates MeCl ca. 30 times slower than elimination of Mel from [MeIr(CO)2l3] (at 93-132 °C in PhCl). 

Kinetic data analyzed using a single-phase exponential model. 

Different theoretical models exist that describe mathematically the adsorption of gases in such a way that an exponential relationship between the amount and the rate of adsorption results. These models are used to obtain from Elovichian type kinetic data information about the energetic reaction parameters. Different models can be used which differ from each other by their physical approach to the reaction process. 

The kinetic traces are obtained from stopped-flow experiments and fit to a single-exponential or to a sum-of-exponentials model. Strictly speaking, one does not know beforehand how many exponentials will be required to describe adequately a particular kinetic curve, so that one has to perform several separate fits, each using a different number of such terms. We therefore need criteria to use in choosing which model best describes the data. The standard ones are the form of (i) the residuals (ii) the autocorrelation function values and (iii) the value (statistical goodness-of-fit parameter). 

Transient-state kinetic data are typically fit with multiple exponentials and not with analytically derived equations. This procedme yields observed rate constants and amplitudes, each of which is typically assigned to one process. These amplitudes can be complex functions of rate constants, extinction coefficients, and intermediate concentrations. It can be difficult to extract meaningfiil parameters from them without the use of a frill model for the reaction and corresponding mathematical analysis. 

Numerical identifiability also becomes a problem with a poorly or inadequately designed experiment. For example, a drug may exhibit multi-exponential kinetics but due to analytical assay limitations or a sampling schedule that stops sampling too early, one or more later phases may not be identifiable. Alternatively, if sampling is started too late, a rapid distribution phase may be missed after bolus administration. In these cases, the model is identifiable but the data are such that all the model components cannot be estimated. Attempting to fit the more complex model to data that do not support such a model may result in optimization problems that either do not truly optimize or result in parameter estimates that are unstable and highly variable. 

In the second test, a number of fluorescent compounds of relatively well known lifetimes in the nanosecond time range (8,9) were used as standards, allowing evaluation of both the instrumental and computational aspects of the measurement. Table I shows the values obtained for 2,3-diphenyloxazole (PPO), anthracene and quinine blsulphate. All chemicals were analytical grade and not further purified before use. Anthracene and PPO were dissolved in cyclohexane, quinine in O.IN 8280 solvents were not degassed. The case of quinine is of interest because of its common use as a standard for fluorescence measurements, despite its complex decay kinetics (10). In agreement with previous work (10) we found satisfactory fits of our deconvolved data to a blexponentlal rather than a single exponential model. 

Thirdly, together with the bimolecular chain termination a monomolecular chain termination is also included in the kinetic models. The latter is controlled by the rate of chain propagation and represents the active center of a radical self-burial act. It has been determined, based on postpolymerization kinetics data (see Chapter 7), that the relaxation function of monomolecular chain termination follows the stretched exponential law and the rate constant has an activating nature with an activation energy typical for a constant of chain propagation and its. scale depends on the molar-volumetric concentration of the... 

Model systems have been devised to investigate electron transfer between cytochrome c and molecular complexes such as [Ru(NH3)6], and kinetic data are consistent with Marcus theory, indicating outer-sphere processes. For electron transfer in both metalloproteins and the model systems, the distance between the metal centres is significantly greater than for transfer between two simple metal complexes, e.g. up to 2500pm. The rate of electron transfer decreases exponentially with increasing distance, r, between the two metal centres (eq. 26.65, where p is a parameter which depends on the molecular environment). 

Compared to reactions in solution, the interpretation of kinetic data at solid-electrolyte interfaces is often difficult because of the heterogenity of suspended particles. In general, the kinetic data recorded (concentration as a function of time) are interpreted by use of a special model like first- and second order kinetics, Elovich s equation or a yrlaw [1]. These models imply some special experimental constraints, e.g., a limited concentration and time range. In Elovich s equation an exponentially de-... 

Under Properties, Specifications, select the base property method. Since these components are liquids, NRTL thermodynamic package is the most convenient fluid package. Install CSTR reactor under Reactors in the model library, and connect inlet and exit streams. Specify the feed stream conditions and composition. Input the reactor specifications double click on the reactor block. The reactor Data Browser opens. Specify an adiabatic reactor and the reactor volume to 4433 liters the value obtained from hand calculations (Figure 5.11). Add the reactions to complete the specifications of the CSTR. Choose the Reactions block in the browser window and then click on Reactions. Click New on the window that appears. A new dialog box opens enter a reaction ID and specify the reaction as Power Law. Then click on Ok. The kinetic data are very important to make Aspen converge. Mainly specifying accurate units for pre-exponential factor A, is very important (see the k value in Figure 5.12). The value MUST be in SI units. 

We can also turn the question around. In chemical kinetics, we need a model to fit the data. This model can be simple, as in first-order reactions where the decay is exponential, or more complicated depending on a complex mechanism. If we do not have a model, our data are just that, data. We could try to fit to a variety of functions, but as there is an infinite number of different functions, that is a pointless exercise. As we have seen in the classical part of this chapter, even for a simple reaction a variety of models are possible, based on dissipative classical dynamics, and we can use these models to try to understand our data. This often involves varying the external parameters, temperature, pH, viscosity, and polarizabihty, but our model should tell us what to expect for such variations for instance, how the rate constant for a reaction depends on those parameters. If our models are quantum mechanical in nature, it is mandatory that we also provide a mechanism for decay, and show how the decay constant or constants depend on external parameters. 

Since the excited state was associated with the dissociation of CH3OH, the irradiation time-dependent excited resonance signal illustrated essentially the kinetics of photocatalyzed dissociation of CH3OH on rutile TiO2(H0). The integrated time-dependent excited resonance signal (Fig. 11.18b) could not be described by a single exponential model, while a fractal-like kinetic model (Eq. 11.7) [164, 165] stimulated the data well ... 

Model fits of the experimental data show that it is also possible to use simplified first-order elementary reaction kinetics for these catalysts to approximate the WGS reaction as a single reversible surface reaction. Furthermore, the fitted values for the pre-exponential coefficients and the activation energies have been evaluated and are not much different from other data available in the open literature. 

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