Isothermal kinetic equation

In fact, all isothermal kinetic equation can be expressed as the following equation,... 

Theoretical formulation of kinetic expressions from specified geometry and/or mechanisms of reaction have often assumed particles to be of a regular, perhaps defined, shape and of uniform size. Equations developed in this way have frequently been found to give a satisfactory representation of observed isothermal kinetic characteristics in many reactions of interest. Other authors have, however, introduced an allowance for particle size distribution [480—482] into kinetic analyses. 

Tests of obedience of isothermal kinetic data to theoretical kinetic equations... 

Many non-isothermal kinetic studies use a linear rate of temperature increase from an initial value, T0, so that the three equations involved can be written... 

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... 

References to a number of other kinetic studies of the decomposition of Ni(HC02)2 have been given [375]. Erofe evet al. [1026] observed that doping altered the rate of reaction of this solid and, from conductivity data, concluded that the initial step involves electron transfer (HCOO- - HCOO +e-). Fox et al. [118], using particles of homogeneous size, showed that both the reaction rate and the shape of a time curves were sensitive to the mean particle diameter. However, since the reported measurements refer to reactions at different temperatures, it is at least possible that some part of the effects described could be temperature effects. Decomposition of nickel formate in oxygen [60] yielded NiO and C02 only the shapes of the a—time curves were comparable in some respects with those for reaction in vacuum and E = 160 15 kJ mole-1. Criado et al. [1031] used the Prout—Tompkins equation [eqn. (9)] in a non-isothermal kinetic analysis of nickel formate decomposition and obtained E = 100 4 kJ mole-1. 

Electrochemical reaction rates are also influenced by substances which, although not involved in the reaction, are readily adsorbed on the electrode surface (reaction products, accidental contaminants, or special additives). Most often this influence comes about when the foreign species I by adsorbing on the electrode partly block the surface, depress the adsorption of reactant species j, and thus lower the reaction rate. On a homogeneous surface and with adsorption following the Langmuir isotherm, a factor 10, will appear in the kinetic equation which is the surface fraction free of foreign species 1 ... 

On inhomogeneous surfaces where adsorption obeys the Temkin isotherm, an exponential factor will appear in the kinetic equation ... 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

THE CONNECTION BETWEEN THE KINETIC EQUATION AND THE BINDING ISOTHERM... 

When one examines the rate of an electrochemical reaction and how it varies with overpotential, it is often found that equations such as (7.150) and (7.150a) (which depend on a Langmuir assumption as to the implied isotherm) are obeyed, and there is no need to modify the kinetic equations to allow for a special isotherm. 

Resin system jEpoxjo [Amine]0 Isothermal (I) or Dynamic (D) DSC Kinetic Equation E kJ/mole In A s Ref. 

This function corresponds to the first order kinetic equation (first term on the right-hand side of the equation) and also reflects the effect of self-acceleration (second term on the right-hand side of the equation) the quantitative measure of this effect is the constant co. Thus the reaction rate is determined by two independent constants co and K. The fit of this equation to experimental data is illustrated in Fig. 2.4. The effect of self-acceleration in anionic polymerization of e-caprolactam was also discussed in other publications, 33 35 The kinetic equation of isothermal polymerization based on Eq. (2.13) can be written as... 

The last factor reflects the role of the non-isothermal effect on the kinetics of the process. The complete kinetic equation for non-isothermal polymerization can be written as... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

Isothermal curves derived from this equation are shown in Fig. 2.19. It is clear that this equation fits the experimental data. A comparison of the kinetic equation (2.48) and the Avrami equation shows93 that any experimental data described by the Avrami equation can be approximated by Eq. (2.48) for any arbitrary set of constants. The divergence of the curves does not exceed 1% in the range 0.2 experimental data (in the isothermal case) can be analyzed by both equations with practically the same reliability. Thus the choice of approximating equation depends on the goal of this procedure if we are interested in physi-... 

The parameters in Eq. (2.59) are usually determined from the condition that some function mean-square deviations between the experimental and calculated curves (the error function). The search for the minimum of the function Nelder-Mead algorithm.103 As an example, Table 2.2 contains results of the calculation of the constants in a self-accelerating kinetic equation used to describe experimental data from anionic-activated e-caprolactam polymerization for different catalyst concentrations. There is good correlation between the results obtained by different methods,as can be seen from Table 2.2. In order to increase the value of the experimental results, measurements have been made at different non-isothermal regimes, in which both the initial temperature and the temperature changes with time were varied. 

The study and control of a chemical process may be accomplished by measuring the concentrations of the reactants and the properties of the end-products. Another way is to measure certain quantities that characterize the conversion process, such as the quantity of heat output in a reaction vessel, the mass of a reactant sample, etc. Taking into consideration the special features of the chemical molding process (transition from liquid to solid and sometimes to an insoluble state), the calorimetric method has obvious advantages both for controlling the process variables and for obtaining quantitative data. Calorimetric measurements give a direct correlation between the transformation rates and heat release. This allows to monitor the reaction rate by observation of the heat release rate. For these purposes, both isothermal and non-isothermal calorimetry may be used. In the first case, the heat output is effectively removed, and isothermal conditions are maintained for the reaction. This method is especially successful when applied to a sample in the form of a thin film of the reactant. The temperature increase under these conditions does not exceed IK, and treatment of the experimental results obtained is simple the experimental data are compared with solutions of the differential kinetic equation. 

The term isothermal was previously used in terms of the model of kinetic equations applied to free motion of the particles between strong collisions [18, p. 126 65], In this particular case the collision integral St(/) of the Bhatnagar-Gross-Krook (BGK) model is found for T (q,t) 7 const. 

As stated in the previous section, the use of a phenomenological kinetic equation derived from Eq. (5.1), for a system that does not verify the required restrictions for its use, may lead to different kinetic expressions when trying to fit experimental results obtained under isothermal and nonisothermal conditions. In particular, it may be observed that different kinetic parameters result by varying the heating rates in nonisothermal experiments. 

The quality of the available kinetic equation to fit experimental data in a wide range of isothermal and scanning rate conditions (the numerical solution exhibits a very high parametric sensitivity on the values of the activation energies). 

It might seem that, for the derivation of kinetic equations describing variations in the amounts of substance eqns. (17) and (18), the equation of state is unnecessary. But this is not so. In the case of a variable reaction volume, it may be necessary to express gas-phase substance concentrations through their amounts, since step rates w are specified as functions of concentrations. For isobaric isothermal processes and ideal gases cg = Ne/ V = PN IN otRT. 

When reactants or intermediates are adsorbed, the rate of the reaction may no longer be related to the concentration by a simple law. This situation is best understood where a reactant is nonspecifically adsorbed in the outer -> Helmholtz plane. The effect of such adsorption on the electrode kinetics is usually termed the -> Frumkin effect. Physical and chemical adsorption on the electrode surface is usually described by means of an -> adsorption isotherm and kinetic equations compatible with various isotherms such as the - Langmuir, -> Temkin, -> Frumkin isotherms are known. 

If adsorption equilibrium is established rapidly and the adsorbed and bulk species remain in equilibrium throughout the reaction, cAads can be expressed in terms of a suitable isotherm. This allows the differential kinetic equation to be integrated. For example, if Henry s law adsorption is presumed to apply [43]... 

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