Integrated rate law described

Integrated rate law describes the concentration as a function of time... 

Besides its qualitative description, radioactive decay has an important quantitative description. Radioactive decay can be described as a first-order reaction, that is, the number of decays is proportional to the number of decaying nuclei present. It is described by the integrated rate law... 

Again plotting concentration versus time using these integrated second-order rate laws gives linear plots only if the reaction is a second-order process. The rate constants can be determined from the slopes. If the concentration-time plots are not linear, then the second-order rate equations do not correctly describe the kinetic behavior. There are integrated rate laws for many different reaction orders. 

For such a second-order reaction, a plot of Ijc against t is linear (Fig. 18.7). The factor 2 multiplying kt in this expression arises from the stoichiometric coefficient 2 for NO2 in the balanced equation for the specific example reaction. For other second-order reactions with different stoichiometric coefficients for the reactant (see the thermal decomposition of ethane described on page 755), we must modify the integrated rate law accordingly. 

However, the rate law could also involve [A][B], but that case will not be described. If the stoichiometry is such that [Aj — [A] = [BJ — [B], the integrated rate law is... 

In Chapter 2, several types of kinetic schemes were examined in detail. While the mathematical apparatus was developed to describe these cases, little was said about other methods used in kinetic studies or about experimental techniques. In this chapter, we will describe some of the methods employed in the study of kinetics that do not make use of the integrated rate laws. In some cases, the exact rate law may be unknown, and some of the experimental techniques do not make use of the classical determination of concentration as a function of time to get data to fit to a rate law. A few of the techniques described in this chapter are particularly useful in such cases. 

The integrated rate law for a reaction describes the relationship between the concentration of a reactant and time. 

Although kinetic methods based on differential laws are more exact and more generally applicable, integrated rate laws have the advantage of being more rapid. In addition, in some cases the integrated rate equations can be used to describe the entire course of a chemical reaction. 

Equation (5.11) is the differential form of the rate law which describes the rate at which A groups are used up. To test a proposed rate law and to evaluate the rate constant it is preferable to work with the integrated form of the rate law. The integration of Eq. (5.11) yields different results, depending on whether the concentrations of A and B are the same or different ... 

In the previous two chapters (Chapters 26 and 27), we showed how kinetic laws describing the rates at which minerals dissolve and precipitate can be integrated into reaction path and reactive transport simulations. The purpose of this chapter is to consider how we can trace the reaction paths that arise when redox reactions proceed according to kinetic rate laws. 

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

First-Order Kinetics, K[A] Unimolecular processes, such as ligand dissociation from a metal center or a simple homolytic or heterolytic cleavage of a single bond, provide a straightforward example of a first-order reaction. The kinetics of this simple scheme, Equation 8.5, is described by a first-order rate law, Equation 8.6, where A stands for reactants, P for products, [A]0 for initial concentration of A, and t for time. The integrated form is shown in Equation 8.7 and a linearized version in Equation 8.8. 

The more usual procedure for estimating % and Ta/ from experimental data taken at different temperatures consists in considering N/ distinct isothermal problems and estimating the relevant values of the rate constants then, from these data and the relationship (3.59) it is possible to estimate koi and a/ Nevertheless, since the law describing the temperature dependence of the rate constants is known, it is possible to estimate directly koi and E. To deal with 9 different isothermal runs, it is only necessary to repeat the integration 9 times for each computational step of the objective function in other words, the dimension Nz of the data can be eliminated by posing in series the 9 sets of data. 

With these additional relationships, one observes that if the rate law is given and the concentrations can be expressed as a function of conversion, then in fact we have as a function ofX and this is all that is needed to evaluate the design equations. One can use either the numerical techniques described in Chapter 2, or, as we shall see in Chapter 4, a table of integrals. 

Linear reaction systems allow the rate laws to be presented in a closed form even if the reaction procedure is complex. But non-linear systems cause extreme difHculties in the integration of even simple equations. Therefore quite a few methods are described in the literature to approximate the solution of the differential equation. Nowadays such iterations are no longer necessary, since the relationship between concentrations can be calculated in an easy way for given parameters. Nevertheless in kinetic analysis two questions are essential ... 

In the introduction to this section a wording was used which is of some importance to chemical process safety knowledge of a reaction rate law which describes the investigated process with sufficient accuracy. Nature is complex, so that the desired process is very rarely the only one to proceed under the conditions chosen for the manufacture of a desired plant product. Normally, numerous reactions take place simultaneously. Based on experience and know-how the development chemist was able only to optimize the process with respect to operational conditions up to an extent that the desired process is favoured. But it remains part of reality that the heat production rate measured and the reaction enthalpy obtained by its integration represent gross values which are formed as the sum of all simultaneously contributing reactions. 

Many chemical reactions in the solid state follow rate laws that are based on the process of nucleation. The active sites have been observed microscopically in some cases, and the phenomenon of nucleation is well established. Although they will not be described in detail, several other processes have nucleation as an integral part of at least the early stages. For example, crystal growth has been successfully modeled by this type of rate law. Condensation of droplets is also a process that involves nucleation. Consequently, kinetics of a wide variety of transformations obey rate laws that have some dependence on a nucleation process. 

For the inner zone, in which both transport and reaction occurs, the differential equations are those of the first stage, but the boundary conditions arc dC Jdr = 0 at r = 0 and Eq. 4.2-11 at the boundary with the outer zone. This mt el corresponds to that set up by Ausman and Watson, to describe the rate of burning of carbon deposited inside a catalyst particle [8]. Analytical integration of this fairly general two-stage model is only possible for a zero-order, first-order or pseudo-first-order rate law, whereby Eq. 42-8 reduces to... 

Mathematical rate laws can be developed to describe the rate at which a reaction process occurs. This rate is typically expressed in terms of the change in the concentration of a particular species taking place in the reaction as a function of time. Often, both differential dcjdt) and integrated [c,(t)] rate laws are useful for answering questions about a chemical reaction process. 

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