Kinetic rate equation, zero-order

In the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

Figure 11.1 illustrates the behavior of Equation 11.6. By the assumption of rapid equilibrium the rate determining step is the unimolecular decomposition. At high substrate composition [S] KM and the rate becomes zero-order in substrate, v = Vmax = k3 [E0], the rate depends only on the initial enzyme concentration, and is at its maximum. We are dealing with saturation kinetics. The most convenient way to test mechanism is to invert Equation 11.6... 

CHEMICAL KINETICS RATE SATURATION MICHAELIS-MENTEN EQUATION ZERO-ORDER REACTIONS ORDER OF REACTION MOLECULARITY... 

For carrier-mediated transport, the rate of movement across a membrane will now be constant, since flux is dependent on the capacity of the membrane carriers and not the mass of the chemical to be transported. These processes are described by zero-order kinetic rate equations of the form ... 

The analysis of the release profiles and the kinetic data indicate two different behaviors and a sudden change between them. In the first behavior, which corresponds to the matrices that release the drug at slow rates, the release was controlled by the fully hydrated gel layer. For these matrices, erosion of the hydrophilic gel structure has shown an important influence on drug release. This is indicated by the better fit of the drug release kinetics to the zero-order equation, the n value of... 

In this respect, the overall mineralization rate of phenol has often been approximated with a zero-order reaction rate (Salaices et al., 2004) as it follows a fairly straight line. For the Fe-assisted PC reaction, however, this approximation cannot be applied given the sharp change of slope in the last part of the photoconversion reaction. Thus, a more complex kinetic rate equation needs to be developed to account for this behavior. 

It is clear that such a rate equation cannot be fitted by a power law kinetic model over the whole range of concentration. Equation (3.13) when compared to power law kinetics shows an effective order of 1 at a low gas phase concentration of A and an effective order of zero at a high gas phase concentration of A. Therefore, even for such an extremely simple system, power law kinetics will only be valid over a certain range of concentrations. Figure (3.1) shows a simple plot for the kinetic rate equation (3.13), where given by ... 

As far as an IEMR is concerned, analytical closed-form solutions are available for limiting first and zero order kinetic rate equations.33... 

The rate equation is first-order in acetone, first-order in hydroxide, but it is independent of (i.e., zero order in) the halogen X2. Moreover, the rate is the same whether X2 is chlorine, bromine, or iodine. These results can only mean that the transition state of the rds contains the elements of acetone and hydroxide, but not of the halogen, which must enter the product in a fast reaction following the rds. Scheme VI satisfies these kinetic requirements. 

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. 

In kinetics, reactions are classified as being first, second, third, etc. order depending on the way the rate of the reaction is related to the concentration terms in the rate equation. If the rate of reaction is apparently independent of concentration, the reaction is said to be of zero order. 

When microorganisms use an organic compound as a sole carbon source, their specific growth rate is a function of chemical concentration and can be described by the Monod kinetic equation. This equation includes a number of empirical constants that depend on the characteristics of the microbes, pH, temperature, and nutrients.54 Depending on the relationship between substrate concentration and rate of bacterial growth, the Monod equation can be reduced to forms in which the rate of degradation is zero order with substrate concentration and first order with cell concentration, or second order with concentration and cell concentration.144... 

Initially, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A and B. However, given that all the reaction stoichiometric coefficients are unity, and the initial reaction mixture has equimolar amounts of A and B, it seems sensible to first try to model the kinetics in terms of the concentration of A. This is because, in this case, the reaction proceeds with the same rate of change of moles for the two reactants. Thus, it could be postulated that the reaction could be zero order, first order or second order in the concentration of A. In principle, there are many other possibilities. Substituting the appropriate kinetic expression into Equation 5.47 and integrating gives the expressions in Table 5.5 ... 

Based on these rate laws, various equations have been developed to describe kinetics of soil chemical processes. As a function of the adsorbent and adsorbate properties, the equations describe mainly first-order, second-order, or zero-order reactions. For example. Sparks and Jardine (1984) studied the kinetics of potassium adsorption on kaolinite, montmorillonite (a smectite mineral), and vermiculite (Fig. 5.3), finding that a single-order reaction describes the data for kaolinite and smectite, while two first-order reactions describe adsorption on vermiculite. 

Referring to reactions in which the reaction velocity is independent of the reactant under consideration. For example, for the reaction A + B C, if the empirical rate expression is v = A [B], the reaction is first order with respect to B but zero order with respect to A. See Chemical Kinetics Rate Saturation Michaelis-Menten Equation... 

Another example of zero-order kinetics is the rate of dissolution of encapsulated solutes restricted in the egress by passage through a small orifice in the capsule. If a soluble salt is added in addition to the encapsulated solute, one obtains an osmotically driven solute release system. See also Order of Reaction Molecularity Michaelis-Menten Equation Eirst-Order Reaction... 

The optical rotation of the mixture approaches zero (a racemic mixture) over time, with apparent first-order kinetics. This observation was supported by the semi-log plot [ln(a°D/ aD) vs time], which is linear (Figure 1). It has been shown that this racemization process does in fact follow a true pseudo-first-order rate equation, the details of which have been described by Eliel.t30 Therefore, these processes can be described by the first-order rate constant associated with them, which reflects precisely the intrinsic rate of racemization. Comparison of the half-lives for racemization under conditions of varying amino acid side chain, base, and solvent is the basis for this new general method. 

Non-linear pharmacokinetics are much less common than linear kinetics. They occur when drug concentrations are sufficiently high to saturate the ability of the liver enzymes to metabolise the drug. This occurs with ethanol, therapeutic concentrations of phenytoin and salicylates, or when high doses of barbiturates are used for cerebral protection. The kinetics of conventional doses of thiopentone are linear. With non-linear pharmacokinetics, the amount of drug eliminated per unit time is constant rather than a constant fraction of the amount in the body, as is the case for the linear situation. Non-linear kinetics are also referred to as zero order or saturation kinetics. The rate of drug decline is governed by the Michaelis-Menton equation ... 

A number of mechanisms proposed to explain the stereospecificity in polymerizations involve complexing of the monomer to a metal prior to addition to the chain (2). Kinetic evidence has shown that such a mechanism does occur with n-BuMgBr, n-BujMg and s-BuMgBr in THF-toluene solution. These polymerizations follow an internal zero-order rate equation. Bateup (1,8) proposed that the mechanism is... 

This type of reaction for which the rate equation can be written according to the stoichiometry is called an elementary reaction. Rate equations for such cases can easily be derived. Many reactions, however, are non-elementary, and consist of a series of elementary reactions. In such cases, we must assume all possible combinations of elementary reactions in order to determine one mechanism that is consistent with the experimental kinetic data. Usually, we can measure only the concentrations ofthe initial reactants and final products, since measurements of the concentrations of intermediate reactions in series are difficult. Thus, rate equations can be derived under assumptions that rates of change in the concentrations of those intermediates are approximately zero (steady-state approximation). An example of such treatment applied to an enzymatic reaction is shown in Section 3.2.2. 

Preliminary kinetic data on the catalyzed hydrogenation of acrylamide using HRuCl(diop)2 generally show a first-order dependence on hydrogen, between a first- and a zero-order on both ruthenium and substrate, and an inverse dependence on added diop at lower substrate concentrations. These dependences are consistent with the mechanism outlined below (Reaction 4) and the corresponding rate law (Equation 5). The less than first-order dependence on ru-... 

There are quite a few situations in which rates of transformation reactions of organic compounds are accelerated by reactive species that do not appear in the overall reaction equation. Such species, generally referred to as catalysts, are continuously regenerated that is, they are not consumed during the reaction. Examples of catalysts that we will discuss in the following chapters include reactive surface sites (Chapter 13), electron transfer mediators (Chapter 14), and, particularly enzymes, in the case of microbial transformations (Chapter 17). Consequently, in these cases the reaction cannot be characterized by a simple reaction order, that is, by a simple power law as used for the reactions discussed so far. Often in such situations, reaction kinetics are found to exhibit a gradual transition from first-order behavior at low compound concentration (the compound sees a constant steady-state concentration of the catalyst) to zero-order (i.e., constant term) behavior at high compound concentration (all reactive species are saturated ) ... 

It is clear from the kinetics that both ethylene and oxygen adsorption are important since both compounds appear in the rate equations with non-zero orders. Moreover, it is well known that ethylene is not adsorbed on pure silver, but that it does adsorb on a surface that is partially covered with oxygen. This implies that ethylene is either adsorbed on top of pre-adsorbed oxygen or on silver sites that are activated by the presence of oxygen (i.e. by formation of surface oxides, or another form of electron transfer or polarization). Consequently, two different mechanisms arise for the formation of ethylene oxide. The (direct) combustion of ethylene is another point of discussion. Although many favour the idea that different oxygen species are involved, others assume the same oxygen species, but different forms of ethylene adsorption. 

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