Differential method of kinetic analysis

Hence, the differential method of kinetic analysis can be applied. 

Differential methods of kinetic analysis can provide better distinguishability amongst the available kinetic expressions, particularly for the sigmoid group of equations (A2 to A4 in Table 3.3.) and for the geometric processes (R2 and R3 in Table 3.3.). 

The approach to be followed in the determination of rates or detailed kinetics of the reaction in a liquid phase between a component of a gas and a component of the liquid is, in principle, the same as that outlined in Chapter 2 for gas-phase reactions on a solid catalyst. In general the experiments are carried out in flow reactors of the integral type. The data may be analyzed by the integral or the differential method of kinetic analysis. The continuity equations for the components, which contain the rate equations, of course depend on the type of reactor used in the experimental study. These continuity equations will be discussed in detail in the appropriate chapters, in particular Chapter 14 on multiphase flow reactors. Consider for the time being, by way of example, a tubular type of reactor with the gas and liquid in a perfectly ordered flow, called plug flow. The steady-state continuity equation for the component A of the gas, written in terms of partial pressure over a volume element dV and neglecting any variation in the total molar flow rate of the gas is as follows ... 

Tdjle 2 Thermal cracking of propane. Rate versus conversion, k-vaiues from the integral and differential method of kinetic analysis... 

Table I Thermal cracking of acetone. Rate coefficients and order by the integral and differential methods of kinetic analysis...

L2.4.1 The Differential Method of Kinetic Analysis L2.4.2 The Integral Method of Kinetic Analysis Coupled Reactions... 

The experimental results may be analyzed in two ways, as mentioned already in Chapter 1—by the differential method of kinetic analysis or by the integral method, which uses the x versus W/Faq data. The results obtained in an integral reactor may be analyzed by the differential method provided the a versus W/Fao curves are differentiated to get the rate, as illustrated by Fig. 2.5-2. Both methods are discussed in the following section. 

When the v experimental errors are normally distributed with zero mean and those associated with the Mh and kh responses (e.g., in the differential method of kinetic analysis r and r j are statistically correlated, the parameters are estimated from the minimization of the following multiresponse objective criterion ... 

From (9.1-1) it follows that the slope of the tangent of the curve Xa versus V/Fao is the rate of reaction of A at the conversion Xa. The rates are shown in Table 9.2.1-2. The kare calculated by both the integral and the differential method of kinetic analysis. 

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 ... 

The data given below are provided by J. H. Raley, F. E. Rust, and W. E. Vaughn J.A.ChS., 70,98 (1948)]. They were obtained at 154.6°C under a 4.2-mmHg partial pressure of nitrogen, which was used to feed the peroxide to the reactor. Determine t he rate coefficient by means of the differential and integral method of kinetic analysis. 

Figure 2.3.a-2 Relation between differential and integral methods of kinetic analysis and differential and integral reactors. 

The integral method of kinetic analysis can be conveniently used when the expression for can be analytically integrated. When the differential method is applied, N, i4 is obtained as the slope of a curve giving (px)hi (Pa)i>m a function of p, VIF, arrived at by measuring the amount of A abmrbed at different gas flow rates. 

Determine a suitable kinetic model by means of both the differential and integral method of kinetic analysis. 

Apply the differential and integral methods of kinetic analysis (see Chapter 2) to determine the rate coefficients and order at the different temperatures. To work out the integral method of kinetic analysis, it is necessary to express pA as a function of x. A rigorous expression would only be possible if all reactions taking place were exactly known. Therefore, undertake an empirical fit of this function. 

However, with the exception of copolymerization of the three- and/or four-membered comonomers, the copolymerization of higher rings is expected to be reversible, such that four additional homo- or cross-depropagation reactions must be added (kinetic Equation 1.44). In such a situation, the traditional methods of kinetic analysis must be put on hold , as a numerical solving of the corresponding differential equations is necessary. Moreover, depending on the selectivity of the active centers, any reversible transfer reactions can interfere to various degrees with the copolymerization process. Thus, the kinetically controlled microstructure of the copolymer may differ substantially from that at equilibrium (cf Section 1.2.4). 

Mottola, H. A. Catalytic and Differential Reaction-Rate Methods of Chemical Analysis, Crit Rev. Anal. Chem. 1974, 4, 229-280. Mottola, H. A. Kinetic Aspects of Analytical Chemistry. Wiley New York, 1988. 

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

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. 

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

The text reviews the methodology of kinetic analysis for simple as well as complex reactions. Attention is focused on the differential and integral methods of kinetic modelling. The statistical testing of the model and the parameter estimates required by the stochastic character of experimental data is described in detail and illustrated by several practical examples. Sequential experimental design procedures for discrimination between rival models and for obtaining parameter estimates with the greatest attainable precision are developed and applied to real cases. 

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