Analysis of Rate Data

With two adjustable constants, you can fit a straight line. With five, you can fit an elephant. With eight, you can fit a running elephant or a cosmological model of the universe.  

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is 

For enzymatic and other heterogeneously catalyzed reactions, there may be competition for active sites. This leads to rate expressions with forms such as 

All the rate constants should be positive so the denominator in this expression will always retard the reaction. The same denominator can be used with Equation (7.3) to model reversible heterogeneous reactions  

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

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In general, a kinetic study begins with the collection of data of concentration versus time of a reactant or product. As will be seen later, this can also be accomplished by determining the time dependence of some variable that is proportional to concentration, such as absorbance or NMR peak intensity. The next step is to fit the concentration-time data to some model that will allow one to determine the rate constant if the data fits the model. 

A zero-order reaction is rare for inorganic reactions in solution but is included for completeness. For the general reaction 

This predicts that a plot of [B] or [B] - [B]q versus t should be linear with a slope of k. 

Strictly speaking, there is no such thing as an irreversible reaction. It is just a system in which the rate constant in the forward direction is much larger than that in the reverse direction. The kinetic analysis of the irreversible system is Just a special case of the reversible system that is described in the next section. 

The problem, in general, is to convert this differential equation to a form with only one concentration variable, either [A] or [B], and then to integrate the equation to obtain the integrated rate law. The choice of the variable to retain will depend on what has been measured experimentally. One of the concentrations can be eliminated by considering the reaction stoichiometry and the initial conditions. The most general conditions are that both A and B are present initially at concentrations [A]o and [B)o, respectively, and that the concentrations at any time are defined as [A] and IB]. 

To arrive at a rate expression that describes intrinsic reaction kinetics and is suitable for engineering design calculations, one must be assured that the kinetic data are free from artifacts that mask intrinsic rates. A variety of criteria have been proposed to guide kinetic analysis and these are thoroughly discussed by Means [D. E. Means, Ind. Eng. Chem. Process Des. Develop., 10 (1971) 541]. 

A lack of significant intraphase diffusion effects (i.e., 17 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143]  

The influence of mass transfer through the film surrounding a spherical catalyst particle can also be examined with a similar expression. Satisfaction of the following inequality demonstrates that interphase mass transfer is not significantly affecting the measured rate  

Criteria have also been developed for evaluating the importance of intraphase and interphase heat transfer on a catalytic reaction. The Anderson criterion for estimating the significance of intraphase temperature gradients is [J. B. Anderson, Chem. Eng. Sci., 18 (1963) 147]  

While the above criteria are useful for diagnosing the effects of transport limitations on reaction rates of heterogeneous catalytic reactions, they require knowledge of many physical characteristics of the reacting system. Experimental properties like effective diffusivity in catalyst pores, heat and mass transfer coefficients at the fluid-particle interface, and the thermal conductivity of the catalyst are needed to utilize Equations (6.5.1) through (6.5.5). However, it is difficult to obtain accurate values of those critical parameters. For example, the diffusional characteristics of a catalyst may vary throughout a pellet because of the compression procedures used to form the final catalyst pellets. The accuracy of the heat transfer coefficient obtained from known correlations is also questionable because of the low flow rates and small particle sizes typically used in laboratory packed bed reactors. 

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The above mentioned studies were in most cases performed with the aim of obtaining relative reactivities or relative adsorption coefficients from competitive data, sometimes also from the combination of these with the data obtained for single reactions. In our investigation of reesterification (97,98), however, a separate analysis of rate data on several reactions provided us with absolute values of rate constants and adsorption coefficients (Table VI). This enabled us to compare the relative reactivities evaluated by means of separately obtained constants with the relative reactivities measured by the method of competitive reactions. The latter were obtained both from integral data by means of the known relation... 

Example provides another example of the analysis of rate data. 

In variable volume systems the dV/dt term is significant. Although equation 3.0.9 is a valid one arrived at by legitimate mathematical operations, its use in the analysis of rate data is extremely limited because of the awkward nature of the equations to which it leads. Equation 3.0.1 is preferred. 

Stoichacmetry and reaction equilibria. Homogeneous reactions kinetics. Mole balances batch, continuous-shn-ed tank and plug flow reactors. Collection and analysis of rate data. Catalytic reaction kinetics and isothermal catalytic radar desttpi. Diffusion effects. 

Analysis of rate data can be made according to this equation after a differentiation with respect to pressure... 

As mentioned in Section 4, the analysis of rate data resulting from unimolecular reactions is considerably easier than the analysis of such data for bimolecular reactions, and the same is true for pseudounimolecular reactions. Kinetic probes currently used to study the micellar pseudophase showing first-order reaction kinetics are almost exclusively compounds undergoing hydrolysis reactions showing in fact pseudofirst-order kinetics. In these cases, water is the second reactant and it is therefore anticipated that these kinetic probes report at least the reduced water concentration (or better water activity in the micellar pseudophase. As for solvatochromic probes, the sensitivity to different aspects of the micellar pseudophase can be different for different hydrolytic probes and as a result, different probes may report different characteristics. Hence, as for solvatochromic probes, the use of a series of hydrolytic probes may provide additional insight. 

TABLE 10. Analysis of rate data for hydrolysis of A-i-butylbenzaldoxime and 2-i-butyl-3-phenyloxaziridine at 24.2 C by use of Bunnett linear free-energy relationships... 

The present chapter is not meant to be exhaustive. Rather, an attempt has been made to introduce the reader to the major concepts and tools used by catalytic reaction engineers. Section 2 gives a review of the most important reactor types. This is deliberately not done in a narrative way, i.e. by describing the physical appearance of chemical reactors. Emphasis is placed on the way mathematical model equations are constructed for each category of reactor. Basically, this boils down to the application of the conservation laws of mass, energy and possibly momentum. Section 7.3 presents an analysis of the effect of the finite rate at which reaction components and/or heat are supplied to or removed from the locus of reaction, i.e. the catalytic site. Finally, the material developed in Sections 7.2 and 7.3 is applied to the design of laboratory reactors and to the analysis of rate data in Section 7.4. 

Propose a generalized rate expression for testing the data. Analysis of rate data by the differential method involves utilizing the entire reaction-rate expression to find reaction order and the rate constant. Since the data have been obtained from a batch reactor, a general rate expression of the following form may be used ... 

It can be seen that, even for the relatively simple kinetic scheme chosen, the ubi(iuitous and inconstant surface termination constants are such as to make (quantitative analysis of rate data extremely difficult if not impossible. 

Rimstidt J. D. and Newcomb W. D. (1993) Measurement and analysis of rate data the rate of reaction of ferric iron with pyrite. Geochim. Cosmochim. Acta 57(9), 1919-1934. 

Collection and Analysis of Rate Data ch 3 Table E5-6.1. DnTERENnAL Reactor Data... 

Method of initial rates In this method of analysis of rate data, the slope of a plot of In(-rj o) versus IhCao wtU be the reaction order. 

Col/ectiort anef Analysis of Rate Data Chap, s... 

In the integral method of analysis of rate data we are looking for the appropriate function of concentration correspondmg to a particular rate law that is linear with time. You should be thoroughly familiar with the methods of obtaining these linear plots for reactions of zero, first, and second order. 

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