Integrated rate, higher order

The CSTR is not an integral reactor. Since the same concentration exists everywhere, and the reactor is operating at steady state, there is only one reaction rate at the average concentration in the tank. Since this concentration is low because of the conversion in the tank, the value for the reaction rate is also low. This is particularly significant for higher-order reactions compared with integral reactor systems. 

The mathematics involved for higher-order reactions become more difficult and such a treatment is beyond the scope of this book. For example, the integrated rate equations for the three types of third-order reactions are given in Table 3.2. 

Expressions similar to equation (17) may easily be derived for various second-, third-, and higher-order reactions. These expressions are readily integrated for all second-order reactions and for many third- and higher-order reactions, yielding (in many cases) relations analogous to equation (18), which define useful concentration-time graphs. The dimensions of the rate constant k for an nth order reaction are (concentration) (time) ... 

Although the formulation and integration of simple rate equations is not particularly troublesome, reflection on the nature of nonintegral and higher order (larger than unity) rate laws indicates much more difficult mathematics, since these are nonlinear in the concentration variables. Together with the exponential dependence of rate, the nonlinear concentration dependence of rate in these cases means that even apparently simple reactions may be difficult to analyze within the context of practical operation. 

Many authors propose alternative mathematical treatments for kinetics equations. Some examples are a general approach based on a matrix formulation of the differential kinetic equations (Berberan-Santos Martinho, 1990) spreadsheets in which rate equations are integrated using the simple Euler approximation (Blickensderfer, 1990) a method for the accurate determination of the first-order rate constant (Borderie, Lavabre, Levy Micheau, 1990) a simplification of half-life methods that provides a fast way of determining reaction orders and rate constants (Eberhart Levin, 1991) a general approach to reversible processes, the special cases of which are shown to be equivalent to basic kinetic equations (Simonyi Mayer, 1985) an equation from which zero-, first- and higher order equations can be derived (Tan, Lindenbaum Meltzer, 1994). 

While integrated rate expressions can be found for second- and higher-order rate laws, in practice this is rarely useful. To see why, consider the five second-order rate laws in Table 4.1. If one monitors the concentration of A, all five rate laws lead to the same general time dependence. In terms of the progress variable (4.7), these rate laws all have the form... 

In contrast to the first-order mechanisms just considered, the integrated rate laws for most multistep second (or higher)-order elementary processes can only be determined numerically, using computer methods. However, in many cases of interest the ideas of preequilibrium and steady state are applicable and the kinetic analysis is much simplified. 

Equation 6.22, the integrated rate expression for a first order reaction, is a statement that the concentration of A diminishes as an exponential in time starting from the initial concentration [A]q. So, if we measure [A] at various times in the course of the reaction and plot that concentration against time, the resulting curve will be qualitatively different from that of a zero order reaction, and as we shall see, different from a higher order reaction. This is shown in Figure 6.7. 

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