Zero order limit

Reaction rate limited (zero-order kinetics). In this case, the biofilm concentration has no effect on reaction rate, and the biodegradation breakthrough curve is linear. 

The rate of a ehemieal reaetion is of a zero order if it is independent of the eoneentrations of the partieipating suhstanees. The rate of reaetion is determined hy sueh limiting faetors as ... 

Integration between the limits of c = c° when r = 0 and c = c when t = t gives the integrated zero-order rate equation. 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

The main limitation of perturbation methods is the assumption that the zero-order wave function is a reasonable approximation to the real wave function, i.e. the perturbation operator is sufficiently small . The poorer the HF wave function describes... 

Schmid s observation of the dependence of the reaction rate on the square of the concentration of nitrous acid was interpreted by Hammett (1940, p. 294) as due to the rate-limiting formation of dinitrogen trioxide, N203. The consequent attack of the amine by N203 was postulated to be faster therefore the concentration of the amine has no influence on the overall rate (zero order with respect to amine). Similarly, Hammett regards the second factor of Schmid s equation for diazotization in the presence of hydrochloric or hydrobromic acid as the result of the formation of nitrosyl halide. 

The only really different case is the azo coupling reaction of nitroethane investigated by Sterba and coworkers (Machacek et al., 1968a, 1968b). With the 4-nitrobenzenediazonium ion the reaction is zero-order with respect to diazonium ion and first-order in both nitroethane and base. Obviously the rate-limiting step is the dissociation of nitroethane the formation of the anion is slower than its subsequent reaction with this diazonium ion. For reactions with diazonium ions of lower reactivity it was found necessary to use the reaction system of Scheme 12-64 with the nitroethane anion as steady state intermediate (Machacek et al., 1968b). 

In the azo coupling reaction of acetoacetanilide (Dobas et al., 1969b) the reaction steps of Schemes 12-71 and 12-72 constitute a steady-state system, i.e., Arx [B] < Ar [HB+] == 2[Ar —NJ] A 2 — 0 with a fast subsequent deprotonation (Scheme 12-73). As with nitroethane, this reaction is general base-catalyzed because the ratedetermining step is the formation of the anion of acetoacetanilide (Scheme 12-71). In contrast to the coupling of nitroethane, however, the addition of the diazonium ion (Scheme 12-72) is rate-limiting. The overall kinetics are therefore between zero-order and first-order with respect to diazonium ion and not strictly independent of [ArNJ ] as in the nitroethane coupling reaction. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

If the reaction order does not change, reactions with n < 1 wiU go to completion in finite time. This is sometimes observed. Solid rocket propellants or fuses used to detonate explosives can bum at an essentially constant rate (a zero-order reaction) until all reactants are consumed. These are multiphase reactions limited by heat transfer and are discussed in Chapter 11. For single phase systems, a zero-order reaction can be expected to slow and become first or second order in the limit of low concentration. 

A limiting case of Monod kinetics has Ks = 0 so that cell growth is zero order with respect to substrate concentration. Rework Example 12.7 for this situation, but do remember to stop cell growth when S = 0. Compare your results for X and p with those of Example 12.7. Make the comparison at the end of the exponential phase. 

A full development of the rate law for the bimolecular reaction of MDI to yield carbodiimide and CO indicates that the reaction should truly be 2nd-order in MDI. This would be observed experimentally under conditions in which MDI is at limiting concentrations. This is not the case for these experimements MDI is present in considerable excess (usually 5.5-6 g of MDI (4.7-5.1 ml) are used in an 8.8 ml vessel). So at least at the early stages of reaction, the carbon dioxide evolution would be expected to display pseudo-zero order kinetics. As the amount of MDI is depleted, then 2nd-order kinetics should be observed. In fact, the asymptotic portion of the 225 C Isotherm can be fitted to a 2nd-order rate law. This kinetic analysis is consistent with a more detailed mechanism for the decomposition, in which 2 molecules of MDI form a cyclic intermediate through a thermally allowed [2+2] cycloaddition, which is formed at steady state concentrations and may then decompose to carbodiimide and carbon dioxide. Isocyanates and other related compounds have been reported to participate in [2 + 2] and [4 + 2] cycloaddition reactions (8.91. 

If a detailed reaction mechanism is available, we can describe the overall behavior of the rate as a function of temperature and concentration. In general it is only of interest to study kinetics far from thermodynamic equilibrium (in the zero conversion limit) and the reaction order is therefore defined as ... 

A kinetic study of the hydrodefluorination of C F H in the presence of EtjSiH indicated a first-order dependence on both [fluoroarene] and [ruthenium precursor] and a zero-order dependence on the concentration of alkylsilane, implying that the rate-limiting step in the catalytic cycle involves activation of the fluoroarene. The regioselectivity for hydrodefluorination of partially fluorinated substrates such as CgFjH has been accounted for by an initial C-H bond activation as shown in the... 

The reaction (Eqn. 5.4-65) takes place in the liquid phase. The molecules are transferred away from the interface to the bulk of the liquid, while reaction takes place simultaneously. Two limiting cases can be envisaged (1) reaction is very fast compared to mass transfer, which means that reaction only takes place in the film, and (2) reaction is very slow compared to mass transfer, and reaction only takes place in the liquid bulk. A convenient dimensionless group, the Hatta number, has been defined, which characterizes the situation compared to the limiting cases. For a reaction that is first order in the gaseous reactant and zero order in the liquid reactant (cm = 1, as = 0), Hatta is ... 

FIG. 28 Normalized steady-state diffusion-limited current vs. UME-interface separation for the reduction of oxygen at an UME approaching an air-water interface with 1-octadecanol monolayer coverage (O)- From top to bottom, the curves correspond to an uncompressed monolayer and surface pressures of 5, 10, 20, 30, 40, and 50 mN m . The solid lines represent the theoretical behavior for reversible transfer in an aerated atmosphere, with zero-order rate constants for oxygen transfer from air to water, h / Q mol cm s of 6.7, 3.7, 3.3, 2.5, 1.8, 1.7, and 1.3. (Reprinted from Ref. 19. Copyright 1998 American Chemical Society.)... 

A useful application of the model is to examine the S02 and 02 concentration profiles in the trickle bed. These are shown for the steady-state conditions used by Haure et al. (1989) in Fig. 25. The equilibrium S02 concentration drops through the bed, but the 02 concentration is constant. In Haure s experiments 02 partial pressure is 16 times the S02 partial pressure. At the catalyst particle surface, however, 02 concentration is much smaller and is only about one-third of the S02 concentration. This explains why 02 transport is rate limiting and why experimentally oxidation appears to be zero-order in S02. 

As noted earlier, perhaps the key determinant of the overall release characteristics from an osmotic pump system is the solubility of the drug. If the solubility of the drug is too low, then the total dose that can be delivered is limited. If the solubility is too high, then the percent of the total dose that can be delivered at a zero-order rate can be relatively small. For example, as shown by Theeuwes [10], the fraction of the total dose (F) that can be delivered at a zero-order rate is given by... 

This mechanism can only be regarded as of limited utility since it does not take into account the zero order dependence on catalyst that is observed under some conditions. More investigation is needed to expand the understanding of the system over a wider range of conditions using a rigorous statistical design to try and determine the extent of interactions between the different reaction parameters. 

The intrinsic rate expressions for these reactions are both first-order in hydrogen and zero-order in acetylene or ethylene. If there are diffusional limitations on the acetylene hydrogenation reaction, the acetylene concentration will go to zero at some point within the core of the catalyst pellet. Beyond this point within the central core of the catalyst, the undesired hydrogenation of ethylene takes place to the exclusion of the acetylene hydrogenation reaction. 

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