Linear rate equation

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

The plot of ( versus q which results from Eqn. 9-9 is a polarization curve this polarization crirve is usually divided into two ranges of polarization as shown in Fig. 9-3 one is a range of polarization where a linear rate equation holds near the equilibrium potential (t) - 0) the other is a range of polarization (the Tafel range) where an exponential rate equation applies at potentials away from the equilibrium potential (ti 0). 

Benson (I) shows a method for solving such a set of linear rate equations. The general solution will be... 

The linear rate equation, eqn. (18), was assumed to hold throughout Sect. 2 because it is the most simple case from a mathematical point of view. Evidently, it is valid in the case of the linear mechanism (Sect. 4.2.1) as it is also in some special cases of a non-linear mechanism (see Table 6 and ref. 6). The kinetic information is contained in the quantity l, to be determined either from the chronoamperogram [eqn. (38), Sect. 2.2.3] or from the chronocoulogram [eqn. (36), Sects. 2.2.2 and 2.2.4], A numerical analysis procedure is generally preferable. The meaning of l is defined in eqn. (34), from which ks is obtained after substituting appropriate values for Dq2 and for (Dq/Dr)1/2 exp (< ) = exp (Z) [so, the potential in this exponential should be referred to the actual standard potential, see Sect. 4.2.3(a)]. 

The explicit expressions in the case of the linear rate equation have... 

Substitution of eqns. (209) into the linear rate equation (198a) leads to the general current—potential relationship... 

After substitution of these expressions into the Laplace transformed linear rate equation (198a), the following relation... 

See also Tables 4 and 9.) Substitution of eqns. (221) into the Laplace transform of the linear rate equation, eqn. (101), leads to... 

Time and temperature correlation. Figure 10 shows that the data are not fitted by a linear rate equation and the plots of Fig. 11 of the weight gain vs. the logarithm of the time show smooth curves of increasing slope. Thus neither a linear nor a logarithmic equation applies to the decay of oxidation rates with time. 

The above probabilities may be grouped in the matrix given by Eq. 3-17). It should be noted that Eq.(2-18) is not satisfied along each row because the one-step transition probabilities pjk depend on time n. This is known as the non-homogeneous case defined in Eqs.(2-19) and (2-20), due to non-linear rate equations, i.e. Eqs.(3-12). 

In Step 1, the hydrated metal ions lose one H2O molecule and form an intermediate complex with a surface site. The fast relaxation associated with Step 1 was ascribed to simultaneous adsorption/desorption of the metal ions on a major portion of the 7-AI2O3 surface sites. In the second step a metal ion-surface complex is formed that results in the release of a proton. This slow relaxation was attributed to the adsorption/desorption of metal ions on the remaining, multiple type sites of the 7-AI2O3 surface that comprise a small fraction of the total surface sites. Yasunaga and Ikeda (1986) characterized the first type of surface sites as strong sites and the multiple type sites as weak sites. Linearized rate equations relating reciprocal relaxation times to the intrinsic rate constants were developed and validated for the two-step reaction mechani.sm. A plot of the linearized equation for Step 2 (the faster... 

In this work the main aspect has been concerned with the problem of electronic energy relaxation in polychro-mophoric ensembles of aromatic horaopolymers in dilute, fluid solution of a "good" solvent. In this morphological situation microscopic EET and trapping along the contour of an expanded and mobile coil must be expected to induce rather complex rate processes, as they proceed in typically low-dimensional, nonuniform, and finite-size disordered matter. A macroscopic transport observable, i.e., excimer fluorescence, must be interpreted, therefore, as an ensemble and configurational average over a convolute of individual disordered dynamical systems in a series of sequential relaxation steps. As a consequence, transient fluorescence profiles should exhibit a more complicated behavior, as it can be modelled, on the other hand, on the basis of linear rate equations and multiexponential reconvolution analysis. 

A separate mass balance equation is written in the form of Section 10.6.2 for each compartment in the model. Thus a total of n mass balance equations must be written and solved for an n compartment model. The details of these equations and their solution are not provided in this chapter. However, it will be noted that absorption, distribution, and elimination rates are written in the same form as in the previous one- and two-compartment models. The absorption rate for instantaneous, zero-order, or first-order absorption is identical to the previous forms for one- and two-com-partment models. Distribution and elimination rates are written as first-order linear rate equations using micro rate constants. So the distribution rate from compartment 1 to compartment n is given by kj Aj, the distribution rate from compartment n back to compartment 1 equals k i A , and the elimination rate from any compartment is written k o A schematic diagram for the generalized n compartment model is illustrated in Figure 10.90. 

The intrinsic nonlinearity of this set of equations does not permit an analytical solution. Lewis et alS3 and Davis110 have proposed an analytical solution to the problem in the case of low feed substrate concentration, that is for a linear rate equation. An iterative numerical solution for the nonlinear problem was fully developed by Waterland et al.47... 

More complicated reactions can be easily treated by the methods outlined in the preceeding sections, that is (a) determine the coupled diffusion-chemical reaction equations, (b) linearize the equations in the concentration fluctuations, (c) solve the linearized rate equations by Fourier-Laplace transforms, (d) solve the dispersion equation... 

Note, though, that the method here is not restricted to linear rate equations. Equation (viii) may now be written as... 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

Then a set of linear rate equations may be written in terms of a matrix of phenomenological coefficients which satisfy the Onsager relation (Onsager, 1931) ... 

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