Kinetic methods differential

Figure 4 Reaction kinetics plot showing the use of a differential method of rate determination of PP-N6-PP-g-AA ternary blend. Source Ref. 47.

Direct application of the differential equation is perhaps the simplest method of obtaining kinetic parameters from non-isothermal observations. However, the Freeman—Carroll difference—differential method [531] has proved reasonably easy to apply and the treatment has been expanded to cover all functions f(a). The methods are discussed in a sequence similar to that used in Sect. 6.2. 

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. 

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 calculation of the secondary electron spectrum was carried out using the method of Seltzer [183]. For the jth orbital of a molecule, the cross-section differential in kinetic energy w of the ejected electron is written as the sum of close and distant collisions. 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

Most systems involve several interconnected feedback loops. Such systems cannot be analyzed seriously without a proper formalism, but their detailed description using differential equations is often too heavy. For these reasons we (as many others before) turned to a logical (or Boolean) description, that is, a description in which variables and functions can take only a limited number of values, typically two (1 and 0). Section II is an updated description of a logical method ( kinetic logic ) whose essential aspects were first presented by Thomas and Thomas and Van Ham.2 A less detailed version of this part can be found in Thomas.3 The present paper puts special emphasis on the fact that for each system the Boolean trajectories and final states can be obtained analytically (i.e.,... 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

Lateral interactions between adsorbed species may make the kinetic parameters a function of coverage. In this case, it would be incorrect to rely on integral methods that depend on the properties of the entire TPD curve. Miller et al. [33] and Nieskens et al. [36] showed that doing so may induce artificial compensation effects in the results. Differential methods, which analyze a part of a trace are - in... 

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

Another technique of the differentiation method is the initial rate measurement. A series of experiments are carried out for different initial concentrations over a short time period (5 to 10% or less conversion). This approach is different from the experimental run discussed in Figure 5.7. Each rate measurement requires a new experiment with a different initial concentration. The initial rate of the reaction is determined from the curve of the concentration vs. time, as shown Figure 5.9a. The log of the initial rate is then plotted against the log of the initial concentration (Figure 5.9b). If the order of the reaction calculated from the concentration-time curve is different from the one determined by initial rate experiments, interference by the reaction products is expected, leading to complex reaction kinetics. 

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 ... 

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

In general the interpretation of the data is somewhat more complicated than for the differential method. Especially for an unknown complicated kinetic functions, the derivation of the correct reaction rate expression RA from experimental results using Equations 5.41 and 5.33 is more cumbersome than fitting Equations 5.30 and 5.33. This is especially true for complex reaction networks, as in the isomerization and cracking reactions of crude oil fractions, where the integral method is very laborious with which to derive individual rate constants. 

Pigs. 18 and 21). As in the case of Ni0(200°), the initial total order is close to zero when NiO(250°) is used as a catalyst and the reaction rate on the fresh sample decreases with time according to the kinetics of order one (74). Kinetics of order one are not followed, however, on regenerated catalysts. Reaction orders were determined in this case by the differential method and were found to vary from 1 (fresh catalyst) to 0.77 (constant activity). Since the initial total order is, in all cases, zero, it was concluded that, as in the case of the same reaction on NiO(200°), the reaction order with respect to time is apparent and results from the inhibition of the catalyst by carbon dioxide, the reaction product. Modification of the apparent order with the runs indicates that regenerated samples of Ni0(250°) are less inhibited than the fresh catalyst. 

Differential methods for kinetic analysis, proposed in the literature, include the following. 

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 methods used in the analysis of non-isothermal kinetic data can be classified as derivative, also referred to as differential methods, based on the use of equation... 

There are essentially two methods of determining the kinetic parameters k and n differential and integral. In the differential method, [A] is plotted as a function of time for a batch reactor or... 

The classical kinetic analytical methods [1-3] are mainly appUed in two versions (1) kinetic catalytic method based on catalytic reactions and (2) kinetic differential method based on the use of systems with simultaneous reactions of a reagent with several mixture components with similar properties. These versions are recommended for enzyme reactions with a view to determining enzymes, inhibitors and substrates. These reactions are highly sensitive and specific their use is without any doubt of particular interest for some systems for the selective and highly sensitive determination of some components of systems [3] to which GC can be applied. 

In this method (96), the kinetic constants are calculated from the TG curve by a differential method. It takes into account also the thermal effects of reactions which result in a deviation of the sample temperature from the programmed values of the linear heating. Starting with the differential equation for the thermal decomposition of a solid,... 

Sestak (43) compared the kinetic results calculated by five different methods for a system corresponding to the dehydration of -CaS04 0.5H2O. The five methods evaluated mathematically were (1) Freeman and Carroll (83) (2) Doyle (84) (3) Coats and Redfern (85) (4) Horowitz and Metzger (88) and (5) Van Krevelen et al. (87). From these calculations it was found that the deviations of computed values oF E did not differ by more than 10%. Thus, all the methods appear to be satisfactory for the calculation of E within the limits of accuracy required. The errors of each method due to the inaccuracy of visual deduction of values from the TG curves were also calculated. These errors, % and e (errors in calculation of E or n, respectively), were as follows (1) Freeman and Carroll method, eE = 4% and e = 12% (2) Horowitz and Metzger method, ee = 2% (when the correct value of n is assumed) (3) Doyle method. eE = 4%. However, the magnitude of this error depends primarily on the position of the point on the TG curve on which the calculations are being performed. In the case of differential methods, me most accurate data are calculated from the medium-steep parts of the curve. For the approximation method, the accuracy depends on the determination of the curve inflection point temperature. 

The second method, the differential method, employs the rate equation in its differential, unintegrated, form. Values of dcjdt are obtained from a plot of c against by taking slopes, and these are directly compared with the rate equation. The main difficulty with this method is that slopes cannot always be obtained very accurately, but in spite of this drawback the method is on the whole the more reliable one and unlike the integration method it does not lead to any particular difficulties when there are complexities in the kinetic behavior. 

As [R] decreases further, the kinetics again approach pseudo-first-order rates (Region VI), but now with respect to R. As [R] ([A] + [B]) (Region VII), a pseudo-first-order rate again applies, and general differential reaction-rate methods have been developed for this situation. There are also differential methods based on measurements of initial reaction-rates, where the kinetics become pseudo-zero-order. 

Some of the early methods that were developed to analyze data to determine kinetic parameters were based on differential methods. This refers to the fact that the methods do not involve attempts to obtain an integrated rate law, but rather a differential form is used directly. Suppose a reaction follows a rate law that can be written as... 

The case of a simple first-order, irreversible reaction was briefly discussed in Section 1.3. In principle, with Eq. 1.3-5, one value of (C, t) suffices to calculate k when is known. In practice, it is necessary to check the value of k for a set of values of (C, r). This method, called the "integrar method, is simpler than the differential method when the kinetic equation flJ-4) can be integrated. When the order of the reaction is unknown, several values for it can be tried. The stoichiometric equation may be a guide for the selection of the values. The value for which k, obtained from Eq. 1.3-4 or Eq. 1.3-5, is found to be independent of the concentration is considered to be the correct order. 

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 ... 

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... 

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