Differential method of analysis

To outline the procedure used in the differential method of analysis, we consider a reaction carried out isoihermaily in a constant-volume batch reactor and (he concentration recorded as a function of time. By combining the mole bal-u/tce with the rate (aw given hy Equation (5-1). we obtain 

After taking the natural logarithm of both sides of Equation (5-6), 

To obtain the derivative —dCf /dt used in this plot, we must differentiate the concentration-time data either numerically or graphically. We describe three methods to determine the derivative from data giving the concentration as a function of time. These methods are  

With this method, disparities in the data are easily seen. Consequently, it is advantageous to use this technique to analyze the data before planning the next set of experiments. As explained in Appendix A.2, the graphical method involves plotting -ACa/A as a function of / and then using equal-area differentiation to obtain An illustrative example is also given in 

In addition to the graphical technique used to differentiate the data, two other methods are commonly used differentiation formulas and polynomial fitting. 

To obtain the derivative used in this plot, we must differentiate the 

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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 ... 

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. 

The differential method of analysis deals directly with the differential rate equation to be tested, evaluating all terms in the equation including the derivative dCJdt, and testing the goodness of fit of the equation with experiment. 

Differential Analysis. Integral analysis provides a straightforward rapid procedure for testing some of the simpler rate expressions. However, the integrated forms of these expressions become unwieldy with more complicated rate expressions. In these situations, the differential method of analysis becomes more convenient. The procedure is closely analogous to the differential method described in Chapter 3. So, by differentiating Eq. 53 we obtain... 

Gordon Campbell (Ref 17) described the differential method of analysis in the investigation of thermal decompn of inorganic oxidants, such as AN... 

Example 5 1 Differential Method of Analysis of Pressure-Time Data Determine the reaction cadet for the gas-phase decomposition oE di-feri-butyi peroxide,... 

Find the reaction rate equation r = kc, using the differential method of analysis. 

In contrast, with the differential method of analysis, only a single experiment need be carried out. However, in this experiment x(t) and s t) are simultaneously measured. As suggested in Fig. 4.19, measurements made at about 2-hour intervals are sufficient from the beginning of the experiment to the beginning of the exponential growth phase. In the range where Monod kinetics are to be characterized in effect, measurements should be repeated every few minutes (this is also the reason why, for example, semicontinuous addition... 

If the differentiation method of analysis is employed for an assumed mechanism, e.g.. 

For the differentiation method of analysis, Equation (13.45) is written in differential form as... 

Analysis The reaction rate data in this example were obtained at steady state, and as a result neither the integral method nor differential method of analysis can be used. One of the purposes of this example is to show how to reason out the form of the rate law and to then use regression to determine the rate law parameters, Once the parameters were obtained, we showed how to linearize the rate law [e.g Equation (E7-4.13) to generate a single plot of all the data. Figure (E7-4.2). 

The differential method of analysis can be used when numerical values of the reaction rate have been obtained at various concentrations and at a constant temperature. The kind of data that are required is represented in Table 6-1. 

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