Graphical analysis of initial rate data

The graphical analysis of initial rate data, which is described extensively in this chapter, is useful, especially while the experiment is still in progress. However, it is important to emphasize that, for definitive results, one should always fit the data with appropriate rate equations for statistical analysis (Cleland, 1979). 

It is useful to note the identity of terms in both equations, because both equations wiU be used for the graphical analysis of initial rate data in Section 12.1.3. 

Tlie usual procedure for the graphical analysis of initial rate data would be to treat each substrate as the varied substrate at different fixed concentrations of another substrate, maintaining a fixed concentration of the third substrate. All such plots represent a family of straight lines with a common intersection point to the left of the i/Uo-axis. 

In the graphical analysis of initial rate data, it is pradent to use all three plots shown in Figs. 3 and 4. The direct plot of versus [A] wiU show directly the influence of substrate concentration on initial rate of reaction. The two linear plots should be used together, because the Lineweaver-Burk plot serves to visualize the influence of low concentrations whereas the Hanes plot serves to visualize the influence of high concentrations of substrates. The third plot, the Eadie-Hofstee plot, is useful in detecting exceptionally bad measurements (Section 3.11). 

The analysis of initial rate data is performed graphically and statistically. 

The problems of parameter estimation and model discrimination by statistical analysis are most easily understood in relation to a specific example. Let us consider a bisubstrale reaction that obeys the rate Eq. (18.50). The initial rate data from Table 1 are presented graphically in Figs. 3 and 4. The reciprocal plot of l/Uo versus 1/A is a family of straight lines which, in this case, may look either parallel or intersecting due to the fact that the crossover point is distant from the vertical axis, because If is much smaller than K, it would be... 

Because the forms of these expressions match, we can see that if we plot [A] (on the y axis) as a function of t (on the x axis), we will get a straight line. We can also see that the slope of that line (yn) must be equal to -k and the y intercept (b) must be equal to [A]q, the initial concentration of reactant A. Equation 11.4 provides us a model of the behavior expected for a system obeying a zero-order rate law. To test this model, we simply need to compare it with data for a particular reaction. So we could measure the concentration of reactant A as a function of time, and then plot [A] versus t. If the plot is linear, we could conclude that we were studying a zero-order reaction. The catalytic destruction of N2O in the presence of gold is an example of this type of kinetics. A graphical analysis of the reaction is shown in Figure 11.6. 

The two major methods used predominantly in the kinetic analysis of isothermal data on solid-catalyzed reactions conducted in plug-flow PBRs are the differential method and the method of initial rates. The integral method is less frequently used either when data are scattered or to avoid numerical or graphical differentiation. Linear and nonlinear regression techniques are widely used in conjunction with these major methods. 

The interpretation of relaxation data is most often performed either with reduced spectral density or the Lipari-Szabo approach. The first is easy to implement as the values of spectral density at discrete frequencies are derived from a linear combinations of relaxation rates, but it does not provide any insight into a physical model of the motion. The second approach provides parameters that are related to the model of the internal motion, but the data analysis requires non-linear optimisation and a selection of a suitable model. A graphical way to relate the two approaches is described by Andrec et al Comparison of calculated parametric curves correlating 7h and Jn values for different Lipari-Szabo models of the internal motion with the experimental values provides a range of parameter values compatible with the data and allows to select a suitable model. The method is particularly useful at the initial stage of the data analysis. 

Chapter 6 deals with the analysis of kinetic data, another subject that receives scant attention in most existing texts. First, various techniques to test the suitability of a given rate equation are developed. This is followed by a discussion of how to estimate values of the unknown parameters in the rate equation. Initially, graphical techniques are used in order to provide a visual basis for the process of data analysis, and to demystify the subject for visual learners . Then, the results of the graphical process are used as a starting point for statistical analysis. The use of non-linear regression to fit kinetic data and to obtain the best values of the unknown kinetic parameters is illustrated. The text explains how non-linear regression can he carried out with a spreadsheet. 

---