Arrhenius rate model

With the importance of the devolatilization process to solid particle combustion and the complexity of the chemical and physical processes involved in devolatilization, a wide variety of models have been developed to describe this process. The simplest models use a single or multiple Arrhenius rates to describe the rate of evolution of volatiles from coal. The single Arrhenius rate model assumes that the devolatilization rate is first-order with respect to the volatile matter remaining in the char [40] ... 

Deposition of TiN by the thermal decomposition of tetrakis(dimethylamido)titanium (TDMAT) in a nitrogen atmosphere (as opposed to ammonia) was characterized by a simple Arrhenius rate expression. Adequate deposition rates and good step coverage were achieved for 3/1 aspect ratio holes, 0.40 micron in size. A reactor model was designed,... 

The rate model contains four adjustable parameters, as the rate constant k and a term in the denominator, Xad, are written using the Arrhenius expression and so require a preexponential term and an activation energy. The equilibrium constant can be calculated from thermodynamic data. The constants depend on the catalyst employed, but some, such as the activation energy, are about the same for many commercial catalysts. Equation (57) is a steady-state model the low velocity of temperature fronts moving through catalyst beds often justifies its use for periodic flow reversal. 

It is also possible to use microcalorimetry to obtain useful information about the kinetic processes of the instability (i.e., aggregation, proteolysis) when thermal irreversibility prevails. Scan rates will often distort the onset behavior of the melting transition that can necessarily impose a shift in the Tm, as discussed further in the following text. The scan rate dependence of the Tm may then be used to determine the activation energy of the instability, provided an Arrhenius kinetic model describes the behavior. 

FONI model (a) unique (b) single hysteresis loop or breaking wave (c) isola (d) isola + hysteresis loop (e) mushroom. With the full Arrhenius rate law and the provision pre-heating or cooling, two additional patterns are found (f) reversed hysteresis loop and (g) reversed hysteresis loop + isola. Also shown are various degenerate loci corresponding to parameter values on the boundaries or special points in the parameter plane (see Fig. 7.5). 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

In the Arrhenius-Eyring model of a chemical reaction which takes place without the intervention of light, the reactant(s) R go over to the product(s) P through a transition state (X) which determines the activation barrier Ea in the rate constant equation... 

The success of the Marcus model is connected to the consistent description of self-exchange reactions and later to ET reactions with non-zero free energy. Using the easily measured free energy of reaction (-AGe) in the PES diagram, gives the Arrhenius rate ... 

When several temperature-dependent rate constants have been determined or at least estimated, the adherence of the decay in the system to Arrhenius behavior can be easily determined. If a plot of these rate constants vs. reciprocal temperature (1/7) produces a linear correlation, the system is adhering to the well-studied Arrhenius kinetic model and some prediction of the rate of decay at any temperature can be made. As detailed in Figure 17, Carstensen s adaptation of data, originally described by Tardif (99), demonstrates the pseudo-first-order decay behavior of the decomposition of ascorbic acid in solid dosage forms at temperatures of 50° C, 60°C, and 70°C (100). Further analysis of the data confirmed that the system adhered closely to Arrhenius behavior as the plot of the rate constants with respect to reciprocal temperature (1/7) showed linearity (Fig. 18). Carsten-sen suggests that it is not always necessary to determine the mechanism of decay if some relevant property of the degradation can be explained as a function of time, and therefore logically quantified and rationally predicted. 

The first-order rate coefficient, k, of this pseudo-elementary process is assumed to vary with temperature according to an Arrhenius law. Model parameters are the stoichiometric coefficients v/ and the Arrhenius parameters of the rate coefficient, k. The estimation of the decomposition rate coefficient, k, requires a knowledge of the feed conversion, which is not directly measurable due to the complexity of analyzing both reactants and reaction products. Thus, a supplementary empirical relationship is needed to relate the feed conversion (conversion of A) to some experimentally accessible variable (Ross and Shu have chosen the yield of C3 and lighter hydrocarbons). It is observed that the rate coefficient, k, is not constant and decreases with increasing conversion. Furthermore, the zero-conversion rate coefficient depends on feed specifications (such as average carbon number, hydrogen content, isoparaffin/normal-paraffin ratio). Stoichiometric coefficients are also correlated with conversion. Of course, it is necessary to write supplementary empirical relationships to account for these effects. 

With this demonstration that the kinetics of the phenolysis reaction are second order in nature, an Arrhenius rate constant model... 

Lignin Sulfonate in Phenol (61.5%) using the Arrhenius Rate Constant Model (15)... 

This value of the activation energy, together with the corrected rate constants for each temperature, was employed to evaluate the pre-exponential constant, k0, in the Arrhenius rate constant model. The value thus derived was... 

As with the continuum model, the particle based threshold line probability must at the macroscopic level recover a specified Arrhenius rate coefficient under equilibrium conditions. Unlike the continuum model, however, the Arrhenius constants a and rj do not appear in the model. Instead, the constant of proportionality, Ai, must be obtained through calibration of the model against rate data by performing test simulations under equilibrium conditions. Hence, with this approach, it is not possible to guarantee that the model will produce a specific temperature dependence for the dissociation rate. This could perhaps be achieved through inclusion of further dependencies of the dissociation probability on the various energies involved in the collision. 

In this section, the GCE DSMC chemistry model is calibrated against the QCT database. The parameters now employed in the modified Arrhenius rate coefficient are those derived by Bose and Candler from the QCT data a = 9.45 x 10" m /s, b = 0.42, = 5.925 x J. 

Of the DSMC chemistry models considered in this article, perhaps the GCE model offers the best blend of convenience, flexibility, and accuracy. This type of model is convenient to use because it employs Arrhenius rate coefficient parameters in its mathematical form. So, provided such data exists for a mechanism of interest, it is relatively easy to perform a DSMC computation. The model also provides flexibility through the parameters a, P, and 7 that allow individual energy modes to be biased in the selection of reacting particles. These parameters can be identified for a particular reaction if experimental or numerical data exists that describe the variation of reaction cross section with collision energy. Finally, the model provides accuracy in that it will reproduce the measured equilibrium rate coefficients under conditions of equilibrium. While these attributes of the GCE model are very positive, and this model appears to be adequate for nonionizing air... 

Both the activity of macrofauna and bacteria are temperature sensitive. It was argued earlier that the temperature dependence led to at least a portion of the observed seasonality of pore-water profiles. In the present model it will be assumed that temperature dependence can be described by Arrhenius rate equations using apparent activation energies appropriate for bacterial metabolic activity (the R term) and macrofaunal ac-... 

NH4 profiles at all stations can be successfully fit by the cylinder model. The production rates used in all cases are those given in Fig. 36. These are modiffed as before to appropriate values at particular temperatures by the Arrhenius rate function [Eq. (6.1)] with an apparent activation energy of 18 kcal/mole (Aller and Yingst, 1980). Estimates of bulk sediment diffusion coefficients, D, at different temperatures were made by multiplying the diffusion coefficients at infinite dilution (Li and Gregory, 1974) by the factor cp, where cp = porosity (Manhiem, 1970 Ler-man, 1978 Krom and Berner, 1980). These give

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Figure 10. Arrhenius plots of oxygen isotope rate constants for select feldspar-salt solution (a) and clay/mica-fluid systems based on the simplified pseudo-first order surface area rate model (Eqn. 96).

Figure 2 shows experimental data points for the catalyzed ABA homopolymer system at different temperatures and fitting curves according to equation 13. This figure also indicates that the reaction rate model is adequate. Rate constants and activation energies are listed in Table 1. It is obvious that the catalyst sodium acetate plays a very marginal role. Arrhenius plots for catalyzed and uncatalyzed ABA homopolyesterification reaction is indicated in Figure 3. Kinetics in systems comprising 80 to 90% of ABA were also studied and evaluated according to equation 13 both for uncatalyzed and catalyzed reactions.

In this model, the rate constant, k, is expressed as a function of the pre-exponential factor, the ideal gas constant, R, temperature, T, and the activation energy, E. However, the Arrhenius temperature model often falls short of explaining the physical behavior of foods, especially of macro-molecular solutions at the temperatures above T. A better description of the physical properties is offered by the Williams-Landel-Ferry (WLF) model, which is an expression relating the change of the property to the T -T difference [37,38]. That is. 

In a kinetic study the activation energy is generally not known a priori, or only with insufficient accuracy. The use of the equivalent reactor volume concept therefore leads to a trial-and-error procedure a value of is guessed and with this value and the measured temperature profile Vp is calculated by graphical or numerical integration. Then, for the rate model chosen, the kinetic constant is derived. This procedure is carried out at several temperature levels and from the temperature dependence of the rate coefficient, expressed by Arrhenius formula, a value of is obtained. If this value is not in accordance with that used in the calculation of Vp the whole procedure has to be repeated with a better approximation for . 

The apparent activation energy for the decomposition can be calculated by applying an initial rate (pseudo-zero-order) Arrhenius kinetic model to the critical early stages of the decomposition (typically where the fractional conversion is less than 0.05), that is ... 

K is the reactivity rate constant, D is a constant, and is the critical extent of curing at which the glassy state is attained. Because switching from a reactivity-controlled reaction takes place gradually, the overall rate constant k (x, T) can be expressed using the Rabinovitch model [13] in terms of the Arrhenius rate constant (k) or reactivity rate constant (k ) and diffusion rate constant (k ) as follows [19, 20] ... 

The equation for k is expressed in a variety of ways, but they all have the same parameters. Arrhenius is credited with defining this relationship (in 1899) and proposing the specific reaction rate model which shows the reaction rate is a function of E and T. Rate constants are plotted at several temperatures. The In k (natural log of k) y axis] is plotted at several temperatures as 1/T [x axis] to obtain the slope. 

A reaction rate model was first used by Tobolsky and Eyring to describe the viscoelastic mechanical properties of rubber-like materials. Zhurkov and Korsukov showed that the same model could be used to account for the degradation of a number of polymers under an applied stress. " They derived Eq. (10) for the time to failure, tf, in which A and are the Arrhenius constants for the fracture process, a is a constant (sometimes called the activation volume), and a is the applied stress. 

A typical example of dynamic optimization in ch ical engineering is the change between steady states in a continuous-stured tank reactor (CSTR) in which the irreversible reaction A B takes place ([21,22]) (Figure 14.4). The reaction is first order and exothermic and follows Arrhenius rate law. The reactor is equipped with a cooling jacket with refrigerant fluid at constant temperature T . To develop model equations, we formulate mass and energy balances. 

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