Linear rate law

If the cross-sectional area of the product remains relatively constant but the product layer cracks during growth and so does not impede the access of the gaseous reactairts aird products to the reacting solid, then a linear rate law is observed... 

The models derived for continuous oxide layers remain valuable when porous oxides are formed they provide a frame of reference against which deviations may be examined and give a basis for understanding the factors governing the location of new oxide. In many cases, however, the experimentally derived rate laws no longer have a unique interpretation. For example, the linear rate law relating the thickness of oxide, x, to the time, t... 

If the PBR is less than unity, the oxide will be non-protective and oxidation will follow a linear rate law, governed by surface reaction kinetics. However, if the PBR is greater than unity, then a protective oxide scale may form and oxidation will follow a reaction rate law governed by the speed of transport of metal or environmental species through the scale. Then the degree of conversion of metal to oxide will be dependent upon the time for which the reaction is allowed to proceed. For a diffusion-controlled process, integration of Pick s First Law of Diffusion with respect to time yields the classic Tammann relationship commonly referred to as the Parabolic Rate Law ... 

We have reviewed today s knowledge of the mechanisms for growth of electrolyte crystals from aqueous solution Convection, diffusion, and adsorption ( ) mechanisms leading to linear rate laws, as well as the surface spiral mechanism (parabolic rate law) and surface nucleation (exponential rate law). All of these mechanisms may be of geochemical importance in different situations. 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

Values of effectiveness factors in washcoat layers with non-uniform thickness around the channel perimeter have been studied by Hayes et al. (2005). However, the applicability of (even the generalized) effectiveness factor approach is quite limited in complex systems with competing reactions, surface deposition of reaction components, non-linear rate laws and under transient operating conditions (e.g. periodically operated NSRC). Typically, the effectiveness factor method can be used for more accurate prediction of CO, H2 and HC oxidation light-off and conversions in DOC. 

Of course, each of the two reactions may proceed via a multi-step mechanism of the types discussed in Sect. 4, e.g. 0= 0, O + e Y, Y + e Z, etc. where O and Y are unstable intermediates. In order to avoid too complex mathematics, only such linear mechanisms will be admitted, so that for each of the two overall reactions a linear rate law can be adopted. 

Equation (6.11) is valid for the initial particle growth. Interface control of the growth kinetics would lead to a linear rate law. Details can be found, for example, in [H. Schmalzried (1981)]. 

See, for example, Chap. 2 in G. Goertzel and N. Tralli, Some Mathematical Methods of Physics, McGraw-Hill, New York, 1960. Because Eq. 4.34 is a set of linear rate laws, although coupled, it is possible to express their solutions as the superposition of solutions of uncoupled (i.e., parallel-reaction) rate laws, as in Eq. 4.35. The number of terms in the superposition will be the same as the number of rate laws (two in the present case). The parameters in Eq. 4.35 are then chosen to make the solutions meet all mathematical conditions imposed by the problem to be solved. 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

It follows from eqn. (189) that, when convection is important, the rate constant for the linear rate law is given by... 

The oxidation of iron at high temperatures, where several iron oxide phases form, obeys the parabolic rate law whereas in CO-CO2mixtures above 900°C, it obeys a linear rate law with the exclusive formation of an FeO layer (18). This result is understandable if one considers the high defect concentration in FeO of approximately 10%, which ensures high diffusion velocity. The linear rate constant - exhibits the following de-... 

Figure 8. Comparison of the regression coefficients for parabolic and linear rate laws for plagioclase dissolution at 25°C and a pH range of 3-8...

Above 500 °C the oxide layer cracks, and above 600 °C, WO3 forms on the blue layer. The WO3 is permeable to oxygen. The WO3 formation rate is determined by the tungsten ion transport to the phase boundary WO2.75/WO3. As long as the inner layer has not yet attained its maximum thickness, the growth follows a mixed parabolic linear rate law. When the maximum thickness is reached, it is strictly linear. 

For short times, the radius of the neck between two solid particles grows as when lattice diffusion dominates (Kingery and Berg, 1955). After a short initial period the rate of decrease in the surface area for coalescing solid particles approaches the linear rate law 12.4) with Zf given by (Friedlander and Wu. 1994)... 

When we introduce a suitable coalescence rate law, the extended Smoluchowski equation (12.1) can, in principle, be solved numerically for n(u. u. r) with the appropriate initial conditions. Using the linear rate law (12.4) for solid-state diffusion, Xiong and Pratsinis... 

We are now in a position to consider the rate of the overall reaction. First, we write a formal rate equation for the kinetics of the reduced reaction network, Fig. 7d, by employing the electrical circuit analogy and the linear rate law analogous to Ohm s law. Thus, the overall rate (overall current) is the ratio of the affinity of the OR and the overall resistance of the reaction network. The overall resistance of the reduced reaction network is... 

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