Kinetics isothermal kinetic rate equation

The simplest isotherm is /if = cf corresponding to R = 1. For this isotherm, the rate equation for external mass transfer, the linear driving force approximation, or reaction kinetics, can be combined with Eq. (16-130) to obtain... 

To applying Eq. (2.47) to non-isothermal problems, it is necessary to generalize it by introducing temperature-dependent constants. The basic approach was proposed by Ziabicki94,95 who developed a quasi-static model of non-isothermal crystallization in the form of a kinetic rate equation ... 

The intraparticle transport effects, both isothermal and nonisothermal, have been analyzed for a multitude of kinetic rate equations and particle geometries. It has been shown that the concentration gradients within the porous particle are usually much more serious than the temperature gradients. Hudgins [17] points out that intraparticle heat effects may not always be negligible in hydrogen-rich reaction systems. The classical experimental test to check for internal resistances in a porous particle is to measure the dependence of the reaction rate on the particle size. Intraparticle effects are absent if no dependence exists. In most cases a porous particle can be considered isothermal, but the absence of internal concentration gradients has to be proven experimentally or by calculation (Chapter 6). 

RATE EQUATIONS COMMONLY USED IN KINETIC ANALYSES OF ISOTHERMAL REACTIONS OF SOLIDS... 

It is usually assumed in the derivation of isothermal rate equations based on geometric reaction models, that interface advance proceeds at constant rate (Chap. 3 Sects. 2 and 3). Much of the early experimental support for this important and widely accepted premise derives from measurements for dehydration reactions in which easily recognizable, large and well-defined nuclei permitted accurate measurement. This simple representation of constant rate of interface advance is, however, not universally applicable and may require modifications for use in the formulation of rate equations for quantitative kinetic analyses. Such modifications include due allowance for the following factors, (i) The rate of initial growth of small nuclei is often less than that ultimately achieved, (ii) Rates of interface advance may vary with crystallographic direction and reactant surface, (iii) The impedance to water vapour escape offered by... 

Experiments at a constant temperature are often carried out to investigate the kinetics of a reaction at a high temperature. The rate coefficient is a constant and the rate equation can be solved relatively easily. By var3dng the temperature of isothermal experiments, the dependence of the rate coefficient on temperature may be obtained. 

For isothermal DSC testing Equations 8.8 and 8.9 are used in data reduction to produce degree of cure and cure rate histories. With these histories, the solution of Equations 8.2 and 8.6 yields the kinetic parameters of interest. 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

In recent years the Coal Research Laboratory has been investigating the kinetics and isotherm behavior of methanol sorption on coal (6, 7, 10) along with the sorption of other vapors on coal (6) and of polar vapors on swelling gels (9, 10). Methanol sorption was shown to be reversible on coal, and its sorption behavior supports the model of coal as a gel or mixture of gels in its physical structure. All indications (I, 6, 7) are that its interaction is with specific and a fixed number of sites for a particular coal sample. Although the sorption of methanol is reversible, coal exhibits sorption behavior which is interpreted in terms of an irreversible swelling of the coal gel upon initial exposure to methanol vapor. As a result of these studies, an isotherm and experimental rate equation for the sorption and desorption were derived that fit the observed data. The isotherm derived for methanol sorption on coal was ... 

The intrinsic kinetics was measured in an isothermal integrated reactor and the reaction rate equations in terms of power function have been established... 

Kinetic models referred to as adsorption models have been proposed, especially for olefin polymerisation with highly active supported Ziegler-Natta catalysts, e.g. MgCl2/ethyl benzoate/TiCU AIR3. These models include reversible processes of adsorption of the monomer (olefin coordination at the transition metal) and adsorption of the activator (complexation via briding bonds formation). There are a variety of kinetic models of this type, most of them considering the actual monomer and activator concentrations at the catalyst surface, m and a respectively, described by Langmuir-Hinshelwood isotherms. It is to be emphasised that M and a must not be the same as the respective bulk concentrations [M] and [A] in solution. Therefore, fractions of surface centres complexed by the monomer and the activator, but not bulk concentrations in solution, are assumed to represent the actual monomer and activator concentrations respectively. This means that the polymerisation rate equation based on the simple polymerisation model should take into account the... 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

Wu and Woo [26] compared the isothermal kinetics of sPS/aPS or sPS/PPE melt crystallized blends (T x = 320°C, tmax = 5 min, Tcj = 238-252°C) with those of neat sPS. Crystallization enthalpies, measured by DSC and fitted to the Avrami equation, provided the kinetic rate constant k and the exponent n. The n value found in pure sPS (2.8) points to a homogeneous nucleation and a three-dimensional pattern of the spherulite growth. In sPS/aPS (75 25 wt%) n is similar (2.7), but it decreases with increase in sPS content, whereas in sPS/PPE n is much lower (2.2) and independent of composition. As the shape of spherul-ites does not change with composition, the decrease in n suggests that the addition of aPS or PPE to sPS makes the nucleation mechanism of the latter more heterogeneous. 

This equation is known as the Fnimkin isotherm. It is clear that the Langmuir isotherm is a special case of the Prumkin isotherm, which can be derived from it by setting r = 0. It can also be seen that, for reasonable values of the parameter r, the exponential term in this equation approaches unity for very small values of 0 and becomes constant when 0 is close to unity. Thus, at extreme values of 0, the Prumkin and the Langmuir isotherms lead to the same dependence of coverage on potential, hence to the same rate equations in electrode kinetics. 

The kinetic relationships describing an ion-exchange process are usually based on a mass balance, a rate equation, an equilibrium isotherm, and a... 

To reversely check the kinetic model, the integral rate equation (for non-isothermal conditions) describing the As release during pyrolysis of CCA treated wood is used in combination with the measured temperature profiles T(t) in order to calculate the corresponding As content of the pyrolysis residues. The calculated arsenic content of the pyrolysis residues is compared with the experimental values labscale and TGA experiments) in the parity plot, presented in Figure 5. 

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