Integration of the Rate Equations

That is, the quantity is the chemical flux in the forward direction and k-, Cz is the flux in the reverse direction, the net rate of change of reactant concentration being their difference. 

As the initial conditions we take Ca = c° and Cz = 0 thus the mass balance relationship is = Ca -f Cz, which is combined with Eq. (3-1) to give 

At equilibrium dcjdt = 0 applying this condition to Eq. (3-2) yields (fc, -I-fc, )cX = k iCA, where Ca is the equilibrium value of Ca- This result is used in Eq. (3-2)  

After separation of variables, Eq. (3-3) is integrated to give Eq. (3-4) as the integrated rate equation. 

Evidently simple first-order behavior is predicted, the reactant concentration decaying exponentially with time toward its equilibrium value. In this case a complicated differential rate equation leads to a simple integrated form. The experi- 

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Pari 1 The integral of the rate equation is solved for the time,... 

The global rate of the process is r = rj + r2. Of all the authors who studied the whole reaction only Fang et al.15 took into account the changes in dielectric constant and in viscosity and the contribution of hydrolysis. Flory s results fit very well with the relation obtained by integration of the rate equation. However, this relation contains parameters of which apparently only 3 are determined experimentally independent of the kinetic study. The other parameters are adjusted in order to obtain a straight line. Such a method obviously makes the linearization easier. 

The values of the rate constants are estimated by fitting equations 1.4a and 1.4b to the concentration versus time data. It should be noted that there are kinetic models that are more complex and integration of the rate equations can only be done numerically. We shall see such models in Chapter 6. An example is given next. Consider the gas phase reaction of NO with 02 (Bellman et al. 1967) ... 

The true residence time is found by integration of the rate equation,... 

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. 

In a VOC-NO mixture containing many different organics, the number of reactions becomes unmanageable for application in models used to describe an air basin or region. Thus the amount of computer time required for numerical integration of the rate equations associated with the thousands of individual species found in ambient air is prohibitive. Furthermore, even as computing power increases, in practice, the kinetics and mechanisms required as input are not all known. 

Differential rate equations such as Eq. 2.38 are often used in integrated form, so that the rate constant can be evaluated simply by measuring concentrations rather than rates at selected times t. Integration of the rate equation, however, may be difficult or may give an unwieldy result. In this book, we are concerned primarily with reaction rates as such, and the reader is therefore referred to texts devoted to chemical kinetics30 for information on integrated rate equations. 

Integration of the rate equation is performed to relate the composition to the reaction time and the size of the equipment. From a rate equation such as... 

However, the reverse process, in going from speed to distance, involves integration of the rate equation (6.2). In chemistry, the concept of rate is central to an understanding of chemical kinetics, in which we have to deal with analogous rate equations which typically involve the rate of change of concentration, rather than the rate of change of distance. For example, in a first-order chemical reaction, where the rate of loss of the reactant is proportional to the concentration of the reactant, the rate equation takes the form ... 

The summation is over all reactions in which species j is involved, i.e. i = 1,2,..., R. The justification for the above expressions for pjj and pjk is confirmed later by the agreement with determinations made on the basis of integration of the rate equations. 

We shall recapitulate the governing equations in the next section and discuss the economic operation in the one following. The results on optimal control are essentially a reinterpretation of the optimal design for the tubular reactor. We shall not attempt a full derivation but hope that the qualitative description will be sufficiently convincing. The isothermal operation of a batch reactor is completely covered by the discussion in Chap. 5 of the integration of the rate equations at constant temperature. The simplest form of nonisothermal operation occurs when the reactor is insulated and the reaction follows an adiabatic path the behavior of the reactor is then entirely similar to that discussed in Chap. 8. 

Integration of the rate equation of pol- nnerization [(Eq.(3)j - using the first order decomposition of the... 

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