The Differential Method of Kinetic Analysis

Consider a volume element of the reaction mixture in which the concentrations have unique values. For an irreversible first-order reaction with constant density transforming A into B, (1.1.2-5) and (1.2.1-1) reduce to 

When the rate coefficient k (h ) is known, (1.2.4.1-1) permits the calculation of the rate for any concentration of the reacting component. Conversely, when the change in concentration is measured as a function of time, (1.2.4.1-1) permits the calculation of the rate coefficient. This method for obtaining k is known as the differential method. In principle, with (1.2.4.1-1), one set Ca i) suffices to calculate / when Cao is known. In practice, it is necessary to check the value of k for a set of values of (Q or rather (r i A 

To determine the order, n, of the reaction A 5 for which a value of 1 was taken in (1.2.4.1-1), a number of values of n is chosen and the rate coefficient k is calculated for the sets (r i A The order leading to a unique value 

Another way of determining partial orders is to carry out a number of experiments in which all but one of the reactants are present in large excess. The partial order with respect to A e.g., is then obtained from 

The slope of the straight line in a log r versus log Ca plot is a. The same procedure is then applied to determine the other orders. 

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

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 ... 

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. 

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 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. 

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...

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). 

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 ... 

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. 

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

Although it would appear that plots of ln[—ln(l — a)] against ln(f — t0) provide the most direct method for the determination of n from experimental a—time data, in practice this approach is notoriously insensitive and errors in t0 exert an important control over the apparent magnitude of n. An alternative possibility is to compare linearity of plots of [—ln(l — a)]1/n against t this has been successful in the kinetic analysis of the decomposition of ammonium perchlorate [268]. Another possibility is through the use of the differential form of eqn. (6)... 

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

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... 

The common methods of investigating the kinetics of explosive reactions are differential thermal analysis, thermogravimetric analysis and differential scanning calorimetry. 

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. 

The microanalytical methods of differential thermal analysis, differential scanning calorimetry, accelerating rate calorimetry, and thermomechanical analysis provide important information about chemical kinetics and thermodynamics but do not provide information about large-scale effects. Although a number of techniques are available for kinetics and heat-of-reaction analysis, a major advantage to heat flow calorimetry is that it better simulates the effects of real process conditions, such as degree of mixing or heat transfer coefficients. 

In order to have theoretical relationships with which experimental data can be compared for analysis it is necessary to obtain solutions to the partial differential equations describing the diffusion-kinetic behaviour of the electrode process. Only a very brief account f the theoretical methods is given here and this is done merely to provide a basis for an appreciation of the problems involved and to point out where detailed treatments can be found. A very lucid introduction to the theoretical methods of dealing with transient electrochemical response has appeared (MacDonald, 1977) which is highly recommended in addition to the classic detailed treatment (Delahay, 1954). Analytical solutions of the partial differential equations are possible only in the most simple cases. In more complex cases either numerical methods are used to solve the equations or they are transformed into finite difference forms and solved by digital simulation. 

The approach is, of course, not restricted to the rate expressions of chemical kinetics, and can be applied to a great variety of differential equations. We have used chemical kinetics here because these are important in chemical analysis, and provide a specific topic to illustrate the method. If one can write down the differential equation of a problem, one can also solve it numerically on the spreadsheet. It can be done the quick (but somewhat crude) explicit way illustrated in section 9.2, or more precisely (but with more initial effort) through the implicit method of sections 9.3 and 9.4. 

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