Kinetic rate equation, parabolic

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

In contrast to the Fe2Al5 layer (see Fig. 1.2), the M0AI4 layer is seen to have relatively even interfaces with both initial phases. In the case of the Mo-saturated aluminium melt, its growth kinetics follows the parabolic law jc2 = 2k t (Fig. 5.14). In the 750-850°C range the temperature dependence of the growth-rate constant, ku is described by the equation ... 

In the rate equations relating a to / for a single particle, the (irst did not involve r. In the second, a function of a was proportional to (/ - / )/r, where / is the time at which the process became rate determining. In the third, a function of a was proportional to (/ — to)/r. Kinetics represented by equations of the second and third of these types are described as linear and parabolic, respectively. It was shown that the kinetic curves of a number of alite and cement pastes, some of which contained added alkali sulphates, could be satisfactorily explained (BIOS). For the cements, diffision became virtually the sole rate-controlling process at values of a varying between about 30% and 60%. This appears to agree broadly with the evidence from apparent energies of activation noted in the previous section. 

In principle, the rate equations for surface reaction kinetics are linear and describe a linearly time-dependent growth of the corrosion layer. However, during this growth the oxygen activity on the surface increases and gradually approaches the value for equilibrium of gas phase and oxide surface. Because of the dependence on ao with a negative exponent, the rate gradually decreases, and several authors have misinterpreted this kinetics as parabolic kinetics (see Sect. G.2.3.2). 

All above conclusions are involved as special cases in the general consequences of the collision theory rate equation (51j III) derived in Sec.7.III. The corresponding consequences from the statistical formulation (67.Ill) of the reaction rate theory were also discussed there. The current interpretations of kinetic isotope effects are based on transition state theory. The correction for proton tunneling is first taken into consideration by BELL et al./155/. More extensive work in this direction has been carried out by CALDIN et al. /I53/. In this treatment estimations of the tunneling correction are made using one-dimensional (parabolic) barrier by neglecting the coupling of the proton motion with other motions of reactants or solvent. 

Figure 2 shows the relation between the square of the weight gain and the oxidation time for specimens oxidized in air at temperamre between I200 C and ISOO-C. For all temperatures, oxidation kinetics were of the parabolic type. Thus, the oxidation behaviour is governed by the parabolic rate equation ... 

Integrated, the rate equation describing parabolic reaction kinetics (Fig. 2) is... 

Like the oxidation of hydrocarbons, the autocatalytic oxidation of polymers is induced by radicals produced by the decomposition of the hydroperoxyl groups. The rate constants of POOH decomposition can be determined from the induction period of polymer-inhibited oxidation, as well as from the kinetics of polymer autoxidation and oxygen uptake. The initial period of polymer oxidation obeys the parabolic equation [12]... 

Surface Spiral Step Control. Many crystals grow faster at small supersaturation than allowed by Equation 7. This lead Frank (17) to suggest that steps may also originate from the presence of a screw dislocation, and that this kind of steps is not destroyed by spreading to the crystal edge, but continues infinitely. The rate law according to this theory is parabolic (7). We shall use the following version of the kinetic equation (10)... 

However, when data from many of the kinetics studies on pesticide-soil interactions were plotted according to the parabolic diffusion equation, initial nonlinearity resulted (Fig. 6.4). This suggested that only at longer times did the reaction process conform to PD. The rate-limiting step for this reaction is diffusion into or out of micropores. 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

Equation (2.101) corresponds to transport-controlled kinetics (cf. Stumm 1990). White and Claassen conclude that after long times in natural water/rock systems parabolic rates tend to become linear. Helgeson et al. (1984) show that feldspar dissolution rates are linear if the feldspar is pretreated to remove ultrafine reactive particles. In other words initial parabolic rates are probably an artifact of sample preparation. It seems likely that, in general, the dissolution or weathering of most silicates in natural water/rock systems obeys zero-order kinetics. 

A property that is particularly important for model formation is the linearity or nonlinearity of the reaction kinetics. Linear reaction rates permit models with parabolic differential equations having linear terms. This in turn allows the derivation of explicit solution formulas which are substantially better suited to the simulation of the sensor dynamics than numerical calculations. Since, for the enzyme elec-... 

It is seen from Fig. 2.31a and b [109] that the sublimation rate of VCI (G-2) diminishes both in the bulk and in the surface layer of PE films. The kinetic curve of the initial VCI sublimation has a linear character. At the same time, all desorption curves of G-2 from the extruded PE films (Fig. 2.31a) display parabolic dependencies of the mjmo = art kind and obey Boltzmann s solution of the diffusion equation in a semi-infinite medium [110]. Therefore, it is possible to anticipate that the VCI desorption rate from the film carrier is limited by its diffusion. At the initial moment of diffusion, the surface concentration of the diffusant in the film is equal to that in the volume, although a concentration gradient is formed with time. So, diffusion of VCI in the films within a wide time and temperature range is described by the relation... 

Figure 9. Schematic of the behavior of the parabolic rate constant as a function of oxygen pressure for the oxidation of copper. The rate increases /I/n = I/S, see Equation 20] until the formation of CuO occurs. At that point the chemical potentials of oxygen are fixed both at the Cu-Cu O and at the CugO—CuO interface, and the kinetics become independent of external oxygen pressure (13).

In both of these equations, x is the film thickness, t is the time of the oxidation, and k and are experimentally determined constants. The constant fep is called the parabolic rate constant. A linear rate is usually found when the film is porous or cracked. The parabolic equation is found when the film forms a coherent, impenetrable layer. As the rate of film growth, dx/dt, diminishes with time for the parabolic rate law, this equation is associated with protective kinetics. The parabolic rate law arises when the reaction is controlled by diffusion. The species with the lowest diffusion coefficient plays the most important role in this case. 

Equation (65) demonstrates that decreasing rates are to be expected in the transition from linear to parabolic kinetics. Therefore, the kinetics of oxidation, sulfidation, and so on cannot be exactly linear, even in the start of reaction, but the rate must decrease. This decreasing linear rate often has been misinterpreted, for example, in... 

The reaction rate varies with time according to a parabolic law [191], The kinetics are determined by the diffusion of oxygen through the Si02 layer. The temperature dependence of oxidation follows the Arrhenius equation. 

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