Integration of rate equations

TABLE 7-9 Integration of Rate Equations of a PFR at Constant Pressure... 

Rate equations like 2.27 and 2.28, obtained from a proposed set of elementary reaction steps, are differential equations. Although for our purposes in this book we shall require only differential rate equations, it is usually more convenient in interpreting raw experimental data to have the equations in integrated form. Methods of integration of rate equations can be found in the literature.34... 

Most elementary reactions involve either one or two reactants. Elementaiy reactions involving three species are infrequent, because the likelihood of simultaneous three-body encounter is small. In closed, well-mixed chemical systems, the integration of rate equations is straightforward. Results of integration for some important rate laws are listed in Table 2.7, which gives the concentration of reactant A as a function of time. First-order reactions are particularly simple the rate constant k has units of s , and its reciprocal value (1/k) provides a measure of a characteristic time for reaction. It is common to speak in terms of the half-life ( 1/2) for reaction, the time required for 50% of the reactant to be consumed. When... 

Chandler, J. P., Hill, D. E., Spivey, H. O. A Program for Efficient Integration of Rate Equations and Least Sqares Fitting of Chemical Reaction Data. Computers Biomed. Res. 5, 515 (1972). 

C.E. Housecroft and E.C. Constable (2002) Chemistry, Prentice Flail, Harlow - Chapter 14 covers first order reaction kinetics with worked examples, and includes mathematical background for the integration of rate equations. 

FIGURE 7.7 Numerical integration of rate equations for reactions 2 of Scheme 7.2, performed with the program Qinetic [89]. The concentration of the intermediates I (benzyl, /S-PPE radical, migration product of reaction 4) coincides with the x-axis. 

Other popular methods for numerical solutions of DEs are the Runge-Kutta methods. They again come in forms of different order, depending on the number of selected points on each sub-interval for which the function is evaluated and averaged. The development of these methods includes quite sophisticated analyses of errors (deviations from the true solutions) which occur with functions of different properties. A major problem in the numerical integration of rate equations is stiffness. A differential equation is called stiff if, for instance, different st s in the process occur on widely different time scales. It is very in dent to compute with time intervals suitable for the steepest part of the progress curve (see Press et al., 1986, chapter 16 and commercial programs recommended on p. 36). 

The integration of the differential equation that describes the rate of change of solute concentration within a plate to the volume flow of mobile phase through it. The integral of this equation will be the equation for the elution curve of a solute through a chromatographic column. 

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 ... 

Those special cases of integration of eq. (1), which have found the most widespread application in the development of rate equations for reactions of solids, are discussed below. 

To obtain the time-dependent concentrations of R and P, we need to integrate this rate equation, which is done simply by separation of variables ... 

Kinetic analysis of the data obtained in differential reactors is straightforward. One may assume that rates arc directly measured for average concentrations between the inlet and the outlet composition. Kinetic analysis of the data produced in integral reactors is more difficult, as balance equations can rarely be solved analytically. The kinetic analysis requires numerical integration of balance equations in combination with non-linear regression techniques and thus it requires the use of computers. 

Integrate the rate equation over a range of conversions when starting with C = 2 lbmol/cuft. 

A second order reaction is catalyzed in a PFR by a porous catalyst made up of small spheres. At the inlet the Thiele modulus is 0O = 10. Integrate the rate equation up to 90% conversion. 

Derivation of rate equations is an integral part of the effective usage of kinetics as a tool. Novel mechanisms must be described by new equations, and famihar ones often need to be modified to account for minor deviations from the expected pattern. The mathematical manipulations involved in deriving initial velocity or isotope exchange-rate laws are in general quite straightforward, but can be tedious. It is the purpose of this entry, therefore, to present the currently available methods with emphasis on the more convenient ones. 

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... 

During the TSR process, the concentration of holes and electrons is determined by the balance between thermal emission and recapture by traps and capture by recombination centers, hi principle, integration of corresponding equations yields ric(t,T) and p t,T) for both isothermal current transients (ICTs) or during irreversible thermal scans. Obviously, the trapping parameters hsted together with the capture rates of carriers in recombination centers determine these concentrations. Measurement of the current density J = exp(/in c + yUpP) will provide trap-spectroscopic information. The experimental techniques employed in an attempt to perform trap level spectroscopy on this basis are known as Isothermal Current Transients (ICTs) [6], TSC [7]. 

We can characterize the oscillations in terms of their size (amplitude) and the period between successive peaks. It is particularly useful to establish how the amplitude and period vary with the reactant concentration. One way of doing this is artificially to hold p constant and then integrate the rate equations until a(t) and b(t) settle down to a steady oscillation. Figure 2.5 shows the stable oscillatory response obtained from eqns (2.2) and (2.3) with the reactant concentration held constant at the value p = 0.01 mol dm-3, inside the range of instability. The concentration of species A varies between a maximum value of 1.36 x 10 4 mol dm- 3 and a minimum of 2.77 x 10 7 mol dm-3. The difference between the maximum and minimum gives the amplitude of the oscillation appropriate to this value of p (and to the particular values of the rate constants used from Table 2.1), 1.36 x 10 4 mol dm-3. The period can easily be read off from the figure as the time between successive maxima tp = 19.0s. Similarly, b t) has a maximum of 1.235 x 10 4 mol dm 3 and a minimum of 6.48 x 10 7 mol dm 3, so the oscillatory amplitude is 1.229 x 10 4moldm 3. 

While no complete mechanism has yet been developed which predicts oscillation in a chlorite oscillator from the integration of a set of rate equations derived from elementary... 

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