Data analysis differential method

Integrating the rate equation is often diffieult for orders greater than 1 or 2. Therefore, the differential method of analysis is used to seareh the form of the rate equation. If a eomplex equation of the type below fits the data, the rate equation is ... 

Figure 3-17. Sohematios of the differential method for data analysis.

Modern instruments capable of obtaining excitation-emission matrices (EEMs) allow use of new data-analysis techniques to resolve overlapped spectra. Resolution techniques such as the ratio method (28) and others (29,30) may provide further differentiation of the components present in the phases separated by solvent extraction. 

Since data are almost invariably taken under isothermal conditions to eliminate the temperature dependence of reaction rate constants, one is primarily concerned with determining the concentration dependence of the rate expression [0(Ct)] and the rate constant at the temperature in question. We will now consider two differential methods that can be used in data analysis. 

The following example illustrates the use of the differential method for the analysis of kinetic data. It also exemplifies some of the problems... 

Techniques for the Analysis of Reaction Rate Data that are Suitable for Use with Either Integral or Differential Methods... 

These concentrations may be used in the various integral and differential methods for the analysis of kinetic data that have been described in previous sections. An example of the use of this approach is given in Illustration 3.5. 

Thermoanalytical methods essentially encompass such techniques that are based entirely on the concept of heating a sample followed by well-defned modified procedures, such as gravimetric analysis, differential analysis and titrimetric analysis. In usual practice, data are generated as a result of continuously recorded curves that may be considered as thermal spectra . These thermal spectra also termed as thermograms, often characterize a single or multicomponent system in terms of ... 

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

Whatever the application, MS-based analyses of FAC effluent will always be faced with the need to support a wide range of buffer components, ranging from variable ionic strength, surfactant levels to required cofactors. In select situations such as indicator analyses, online methods may be appropriate but it is clear that the insertion of an intermediate LC step offers sigmficantly improved performance. This changes the nature of the data analysis, from the detection of sigmoidal breakthrough curves to peak detection and differential analysis across multiple fractions. 

There are two procedures for analyzing kinetic data, the integral and the differential methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation, predict that the plot of a certain concentration function versus time... 

In the differential method of analysis we test the fit of the rate expression to the data directly and without any integration. However, since the rate expression is a differential equation, we must first find llV) dNldt) from the data before attempting the fitting procedure. 

Each of these methods has both merits and demerits. For example, the integration method can easily be used to test several well-known specific mechanisms. In more complicated cases the differential method may be useful, but this requires more data points. Analysis by the integration method can generally be summarized as follows. 

Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability. The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values. Three out of seven independently determined factor effects serve for... 

The differential method of analysis of kinetic data deals directly with the differential rate of reaction. A mecha-... 

Such a method of kinetic analysis is termed the differential method since the residual sum of squares is based on rates. The required differentiation of XA versus W/FA0 data can be a source of errors, however. To avoid this, the same set of data can be analyzed by the so-called integral method, which consists of minimizing a residual sum of squares based on the directly observed conversions ... 

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 ... 

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