Parabolic rate equation

A reduction in the oxidation rate due to a change from the thermal diffusion of the ions to electric field -induced transport through the film. The metal oxidation becomes controlled by a combination of both the logarithmic and parabolic rate equations as a result. 

When logarithmic and parabolic oxidation rates occur simultaneously, the metal oxidation may obey cubic rate law. Logarithmic and parabolic rate equations as a function of time may be distinguished by an equation ... 

Table El 1.4 Weight Gain of Cr for Different Chromium Contents Calculated Using Parabolic Rate Equation Ci Parabolic Oxidation Constant kp...

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 ... 

Internal penetration or attack can also be defined by parabolic rate equation. This term describes the formation of precipitates (both inter- and intragranular) by the interaction of one or more constituents of the atmosphere with the substrate. The reactive species of the atmosphere dissolves in the substrate if the gas pressure is helow the dissociation pressiuje of the compounds formed by the reactive species with components of the substrate. The reactive species of the atmosphere diffuse into the substrate and form the most stable precipitate possible. Internal penetration... 

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

Because the rate law in this form is a quadratic equation, the rate law is known as the parabolic rate law. When we solve this equation for the thickness of the product layer, x, we obtain... 

Among the theories proposed, essentially two main mechanisms can be distinguished these are that the rate-determining step is a transport step (e.g., a transport of a reactant or a weathering product through a layer of the surface of the mineral) or that the dissolution reaction is controlled by a surface reaction. The rate equation corresponding to a transport-controlled reaction is known as the parabolic rate law when... 

The quadratic rate equation [Eq. (1)] of the continuum theory arises because it implicitly assumed the parabolic dependence of the free energy profile on the solvent coordinate q. One of the consequences of this quadratic equation is the generation of a maximum in the dependence of the rate of reaction on the free energy of reaction and also in current density-overpotential dependence. 

Because of the appearance of equation 65, B is called the parabolic rate constant. This limiting case is the diffusion-controlled oxidation regime that occurs when oxidant availability at the Si-Si02 interface is limited by transport through the oxide (thick-oxide case). 

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... 

Chien and Clayton (1980) compared several equations for describing P04 release from soils and found that the Elovich equation [Eq. (2.49)] was best based on the highest values of the simple correlation coefficient (r2) and the lowest SE. The two-constant rate equation also described the data satisfactorily. The parabolic diffusion equation was judged unsatisfactory due to low r2 and high SE values. 

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. 

When diffusion is considered as the rate-determining step, the parabolic diffusion equation can be applied (Zimens 1945 Boyd et al. 1947 Crank 1956 Chute and Quirk 1967 Wollast 1967 Jardine and Sparks 1984 Sparks 1999). 

Fig. 2 Schematic representation of a minimal self-replicating system rate equation for parabolic and exponential growth. (Reproduced from [33])...

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. 

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... 

---