Integral methods reversible reactions

Some systems may show stiff properties, especially those for oxidations. Here the system of differential equations to be integrated are not stiff . Even at calculated runaway temperature, ordinary integration methods can be used. The reason is that equilibrium seems to moderate the extent of the runaway temperature for the reversible reaction. 

One advantage of the initial rate method is that complex rate functions that may be extremely difficult to integrate can be handled in a convenient manner. Moreover, if one uses initial reaction rates, the reverse reactions can be neglected and attention can be focused solely on the reaction rate function for the forward reaction. More complex rate functions may be tested by the choice of appropriate coordinates for plotting the initial rate data. For example, a reaction rate function of the form... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

The generalized physical property approach discussed in Section 3.3.3.2 may be used together with one of the differential or integral methods, which are appropriate for use with reversible reactions. In this case the extent of reaction per unit volume at time t is given in terms of equation 3.3.50 as... 

Reversible Reactions in General. For orders other than one or two, integration of the rate equation becomes cumbersome. So if Eq. 54 or 56 is not able to fit the data, then the search for an adequate rate equation is best done by the differential method. 

Initial Rate Method. Using integrated equations like Eqs. (2.5), (2.6), or (2.7) to directly determine a rate law and rate constants is risky. This is particularly true if secondary or reverse reactions are important in equations like (2.5) and (2.6). One sound option is to establish these equations directly using initial rates (Skopp, 1986). 

In many systems, slow reactions occur to an appreciable extent in the presence of fast, reversible reactions. Given extents of reaction measured experimentally, or computed by integration of a kinetic model, it is desired to compute compositions of a system. This has been accomplished using EQUILK (19) given specifications for extents or temperature approaches for less than R reactions, or the moles of some species in the product. The RAND Method has been extended to permit these specifications, as well (38). In principle, these specifications are easily added to the mass balances for any nonlinear programming algorithm. 

This method integrally employs the quasi-steady state assumptions to relate the concentrations of H, OH and O in the overall radical pool, and can be applied to either fuel-rich or fuel-lean flames. Concentrations of HO2 were also calculated using the quasi-steady state condition, but because these were mostly much smaller than the other radical concentrations they were considered in the same manner as OH and O in the simpler method. Both methods lead to similar results for the low temperature, fuel-rich flames considered at present, indicating that the reverse reactions other than (—i) and (—iii) are relatively unimportant over most of these reaction zones. Three internally consistent sets of rate coefficients on which the more refined treatments may be based are given... 

Example 4.2 applied the method of false transients to a CSTR to find the steady-state output. A set of algebraic equations was converted to a set of ODEs. Chapter 16 shows how the method can be applied to PDEs by converting them to sets of ODEs. The method of false transients can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. Formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. Irreversible reactions go to completion. Reversible reactions reach equilibrium concentrations. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

The procedure outlined above can be used to determine all the exponents 2, 3,..., and the rate constant can be evaluated. The advantage of this method is that complex rate equations, which may be difficult to integrate, can be handled in a convenient manner. Also, the reverse reaction can be completely neglected, provided that initial velocities are actually measured or are obtained by an appropriate extrapolation. For reactions having a simple rate law, i.e., first order, second order, etc., the methods discussed previously are more precise. 

The irreversible and reversible complex or multiple reactions have a different behavior and as such cannot be solved by simple integral methods. In most cases, numerical methods are employed. These reactions can occnr in series, parallel, or a combination of both. The goal is to determine the rate constants for reactions of any order. Although the order of these reactions is not integer, we can assnme that it is entire in the different steps. In such cases, an analytical solution can be obtained. The most complex solutions of generic order will not be studied in this chapter. Consider the three cases as follows. 

Chapter 2 covers the basic principles of chemical kinetics and catalysis and gives a brief introduction on classification and types of chemical reactors. Differential and integral methods of analysis of rate equations for different types of reactions—irreversible and reversible reactions, autocatalytic reactions, elementary and non-elementary reactions, and series and parallel reactions are discussed in detail. Development of rate equations for solid catalysed reactions and enzyme catalysed biochemical reactions are presented. Methods for estimation of kinetic parameters from batch reactor data are explained with a number of illustrative examples and solved problems. 

The forward reaction is second order, and the reverse reaction is first order with respect to B and first order with respect to C. Write a computer program using Euler s method to integrate the rate differential equations for the case that the initial concentration of A is nonzero and those of B and C are zero. ... 

When the electrode reaction (2.30) is electrochemically reversible, (2.37) and (2.38) are combined with the Nemst equation (1.8) yielding an integral eqnation that relates the current with time and the electrode potential. The nnmerical solntion derived by the step function method [52] is given by the following recursive formulae ... 

The principle underlying the method is that the ratio of the times at which two specific conversions occur depends solely on the order of the reaction. The method is applicable to single irreversible reactions of any order—positive or negative, integral or fractional—as well as to reversible first-order reactions. 

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