Reaction rate, parabolic

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 ... 

In some circumstances, the reaction rates may not be exactly parabolic, and even initially parabolic rates may be influenced by changes within the oxide scale with time. As an oxide scale grows, the build-up of inherent growth stresses, externally applied strains and chemical changes to either oxide scale or metal may all compromise the initial protection offered by the scale, leading to scale breakdown and ultimately partial or complete loss of protection paralinear, or linear kinetics may ensue. In other circumstances, as will be seen later in this chapter, very small additions of contaminants to... 

Sulphur Trioxide (SO2 -I- O2) Linear reaction rates are observed due to phase boundary control by adsorption of the reactant, SO3. Maximum rates of reaction occur at a SO2/O2 ratio of 2 1 where the SO3 partial pressure is also at a maximum. With increasing 02 S02 ratio the kinetics change from linear to parabolic and ultimately, of course, approach the behaviour of the Ni/NiO system. At constant gas composition and pressure, the reaction also reaches a maximum with increasing temperature due to the decreasing SO3 partial pressure with increasing temperature, so that NiS04 formation is no longer possible and the reaction rate falls. 

The fourth explanation for non-linear kinetics differs from the previous three in that it concerns the composition of the solution rather than any intrinsic property of the solid reactants or products. Changing solution composition can produce apparent or true parabolic dissolution kinetics either through the influence of changing pH and COj equilibria, or through the effect of chemical affinity and the reverse reaction rate. These phenomena have been discussed in detail by Helgeson and Murphy (V7) and Helgeson and others ( 1 8). 

Fig. 8.12 Parabolic reaction rate constant for the formation of LijSb under three different conditions at the opposite side (a) Ou corresponds to that of the equilibrium LijSb/LijSb, (b) Ou = 10 , (c) Ou = 1.

The reasons for the effect of pH on the catalytic properties of enzymes are numerous and will not be discussed here. For most enzymes, however, there is a pH at which they are optimally effective changing the pH to lower (more acidic) levels or to higher (more basic) levels will decrease the overall rate at which the associated chemical reaction occurs. In the region of the optimum pH, the reaction rate vs. pH response surface can usually be approximated reasonably well by a second-order, parabolic relationship. 

The parabolic relationship between reaction rate and uncoded pH is obtained by expansion of the coded model. 

Experimental data (Figure 4.2) for the dissociative electron transfer between radical anions and the carbon-halogen bond in alkyl halides indicates a linear relationship between log(k ) and Ed over a wide range of reaction rates [5, 9]. Very fast reactions become controlled by the rate of diffusion of two species towards each other, when every close encounter gives rise to electron transfer. A parabolic... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

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

The bar over the diffusivity term indicates the product layer average. Ajuao is equal to the standard value of the formation Gibbs energy of the spinel, AGAB2oa. One finds from Eqn. (6.29) that the (parabolic) reaction rate constant (A 2 = 2-kt) is... 

A and B in the A/AmB /B reaction couple. The (parabolic) reaction rate constant k (if local thermodynamic equilibrium prevails throughout the couple) conforms to Eqn. (6.32) if we disregard stoichiometric factors. The pertinent rate constant is then... 

Since this driving force is proportional to A "1, it again leads to a parabolic rate law. The AB formation rate is always decreased compared to a stress-free reaction as long as the layer adheres and does not form cracks. However, if the evolving stress energy contained in the A substrate is also taken into account, the overall stress energy depends on the thickness of the reaction layer, which invalidates the parabolic growth and slows down the reaction rate. In principle, this can stop the reaction before A or B are consumed. 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

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