Isotherms rates

TABLE 7-4 Some Isothermal Rate Equations and Their Integrals... 

The most accurate method of measuring the influence of temperature on reaction rate is to separate the variables by first determining isothermal rate curves at a series of different temperatures and expressing each set of observations in the form... 

Relationship between reaction order, n, and the value of a corresponding to maximum rate, a, during non-isothermal rate processes... 

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... 

From microscopic measurements of the rates of nucleation and of growth of particles of barium metal product, Wischin [201] observed that the number of nuclei present increased as the third power (—2.5—3.5) of time and that the isothermal rate of radial growth of visible nuclei was constant. During the early stages of reaction, the acceleratory region of the a—time plot obeyed the power law [eqn. (2)] with 6 temperature coefficients of these processes were used by Wischin [201]... 

C.G. Vayenas, C.Georgakis, J. Michaels, and J. Tormo, The role of PtOx in the isothermal rate and oxygen activity oscillations of the Ethylene Oxidation on Pt, J. 

Taking into account the concentration invariant CA+CB+CD= CA0, i.e. that there is no change in the total number of moles, the integrated forms of the isothermal rate equations are... 

In this report we present NEXAFS results for the kinetics of ethylidyne formation. Previous data is scarce and comes mostly from thermal desorption (TDS) experiments (2). The only reported study of isothermal rates of reactions for this system was done by Ogle et. al. using secondary ion mass spectrometry (SIMS) (10). 

For exothermic reaction, heat is released and particles are hotter than the surrounding fluid, hence the nonisothermal rate is always higher than the isothermal rate as measured by the bulk stream conditions. However, for endothermic reactions the nonisothermal rate is lower than the isothermal rate because the particle is cooler than the surrounding fluid. 

According to some authors [89,92,140,150,152,157,158], the isothermal rate is in reality expressed by Equation 2.1. The physical meaning of the term (da/3T)l was questioned by Simmons and Wendlant [159], Hill [160], as well as Gorbatchev and Lovinenko [161]. They affirmed that (9a/dT)t has to be zero because if one fixes the time, the positions of all the particles in the system will be fixed, making a to be constant. By using thermodynamics, Kratochvil and Sestak [157] showed that (9a/97 ), exists and is nonzero. 

The least resolved measurement is determination of the isothermal rate constant k(T), where T is the isothermal temperature. Although conceptually simple, such measurements are often exceedingly difficult to perform for activated process without experimental artifact (contamination) because they require high pressures to achieve isothermal conditions. For dissociative adsorption, k(T) = kcol (T) [S (Tg = TS = T)), where kcol(T) is simply the collision rate with the surface and is readily obtainable from kinetic theory and Tg and T, are the gas and surface temperatures, respectively [107]. (S ) refers to thermal averaging. A simple Arrhenius treatment gives the effective activation energy Ea for the kinetic rate as... 

It is easy to conclude that activated CH4 dissociation on Ni(100) (and other transition metals) is dominated by direct rather than precursor-mediated processes under molecular beam conditions because of the strong dependence of S on Et and Tv. However, it is not as easy to decide in thermal bulb experiments and there has been considerable controversy over which dominates the thermal CH4 dissociation [59,285,286]. This is especially true since both lattice coupling in direct dissociation and a precursor-mediated dissociation with Ec>Ed (see Section 2.3.2) can cause k(Tg = 300K, Ts) to increase with Ts. One way to distinguish between these two possibilities is to compare isothermal rates k(T = Ts) with non-isothermal... 

Concerning internal temperature gradients, a criterion which guarantees that the deviation of the observed volumetric rate from the isothermal rate is less than 5% is given in an analogous way by [11] ... 

The isothermal kinetics of decomposition were complex, with at least two overlapping processes taking place. The shapes of the peaks indicated that both processes were initially acceleratory, and then deceleratory. The isothermal rate was assumed to be made up of weighted contributions from individual processes which could be described by the Avrami-Erofeev equation, with various values of n. 

Isothermal reactions A set of isothermal rate constants, measured from experiments made at intervals across the widest accessible temperature range, are frequently used to calculate Ea. The magnitude is often interpreted to identify the energy barrier that must be surmounted in the rate-limiting step or to provide other mechanistic information. Care must be taken to ensure that the data refer to a single-rate process of interest and that rates are not influenced by other secondary controls or secondary reactions. 

PtOx) in the Isothermal Rate Oscillations of Ethylene Oxidation on Platinum. J. Catal. 67, 348-361... 

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