Linear reaction laws

The solute concentrations very close to the interface, and are assumed to be in equiUbrium, in the absence of any slow interfacial reaction. According to the linear distribution law, Cg. = thus from equation 14 the mass-transfer flux can be expressed in terms of an overall... 

All reactions in which charge is transported through a film of reaction product on the metal surface —the film may or may not be rate determining (e.g. parabolic, logarithmic, asymptotic, etc. or linear growth laws, respectively). 

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 ... 

The rate of chemical attack will depend on the concentration according to the order of the reaction (i.e. in a zero-order reaction the rate is independent of concentration, in a first-order reaction the rate depends linearly on concentration, and in second-order reaction the rate depends on the square of concentration). Increasing the concentration, therefore, provides a means of acceleration. Remember, however, that chemical attack on plastics is a liquid-solid and not a liquid-liquid reaction, such that the reaction laws only hold if there is free movement of all chemical species with no limitations due to diffusion or transport and no barrier layers. Since this is rarely the case, temperature is preferred as a means of acceleration. 

Figure 1-13 Parabolic versus linear reaction rate law 55...

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. 

For reactions of the type A + B = AB (or a+P = y), the situation is different. If one has a linear reaction geometry and the y product forms at different times and locations on the A/B interface, the patches of y eventually merge by fast lateral (interface) transport. Eventually, a full y layer is formed between a and / . At first, this layer has a non-uniform thickness (Fig. 6-4). In Chapter 11 we will show, however, that the uneven a/y and y/p interfaces are morphologically stable and become smooth during further growth. This leads to constant boundary conditions for the y formation after some time of reaction and eventually results in a parabolic rate law, as will be discussed later. 

Another important test of the accuracy of the superposition approximation is the diffusion-controlled A + B — 0 reaction. For the first time it was computer-simulated by Toussaint and Wilczek [27]. They confirmed existence of new asymptotic reaction laws but did not test different approximations used in the diffusion-controlled theories. Their findings were used in [28] to discuss divergence in the linear and the superposition approximations. Since analytical calculations [28] were performed for other sets of parameters as used in [27], their comparison was only qualitative. It was Schnorer et al. [29] who first performed detailed study of the applicability of the superposition approximation. 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

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. 

For an overall reaction with / number of intermediate reactions, the linear phenomenological law is valid, if every elementary reaction satisfies the condition A/RT 1, and the intermediate reactions are fast and hence a steady state is reached. 

Linear growth laws (curve (a). Figure 39) are observed for reactions in which the film does not remain coherent, but breaks away from the metal, leading to porosity to the gaseous reactant. The rate is simply the rate of the surface reaction. 

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... 

From Fig. 16 it is seen that the mass gain of Ni30Al is much lower than in the case of Ni36Al apparently due to the lower A1 content. Also there is no incubation period and the mass gain follows an approximately linear rate law, nearly independent of the H2S concentration. Again, however, the reaction rate drops off wherever the H2S flow is switched off (Fig. 17). 

Bronsted and Pedersen [20] indicated that the rate constant for proton transfer from acid to a base cannot continue to increase in accord with a linear Bronsted law but must be limited by an encounter rate. This prediction was confirmed by Eigen s school [21] who showed that changed from 1 to zero as the p/f of the donor acid fell below that of the acceptor base (Fig. 5). Eigen [21] considered the following scheme (sometimes called the Eigen mechanism) for proton transfer from HX to Y where reactions in brackets occur in the encounter complex (Eqn. 28). The overall rate constants are given in Eqns. 29 and 30. 

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