First-order reaction graphical methods

From the design equation for a first-order reaction, eqn. (11), it follows that the reaction time is equal to the area under the curve of l/fe(l — Xa) plotted against Xa- This integral may be obtained graphically by counting squares or by a numerical method. 

Finite difference method digital simulation (see Chapter 1.2 in this volume) has been performed for many different reaction mechanisms, and normalized absorbance working curves have been presented for not only current and charge but also absorbance of the various species assumed by the mechanism [58, 59). For first-order or pseudo-first-order reactions, a little-used method for graphical analysis of the data is to plot the absorption-time transients according to Eq. (32) [40, 60]. 

For determining the rate constants, we use similar methodology as presented previously the integral or differential methods. The differential method is frequently used and easily visualized in the graphic solution after transforming the rate equation. For example, for a monomolecular, irreversible, and first-order reaction, the rate is expressed in Equation 10.22. It was deduced assuming that the reaction rate is the limiting step and both reactants and products adsorbed. Thus,... 

Graphic Method A plot of the data can be used to ascertain the order. If a plot of concentration versus time yields a straight line, the reaction is zero order. A straight line from the plot of logic/ - x) versus time is first order and second order if the plot of 1 /(a - xf versus time is a straight line (where the initial concentrations are equal). 

This graphical method of determining a rate constant, illustrated in Worked Example 12.6, is an alternative to the method of initial rates used in Worked Example 12.3. It should be emphasized, however, that a plot of ln [A] versus time will give a straight line only if the reaction is first order in A. Indeed, a good way of testing whether a reaction is first order is to examine the appearance of such a plot. 

In the graphical method, if the plot of In c versus t is a straight line the reaction is first-order. Similarly, the integrated expression for the second-order reaction can be utilised graphically to ascertain if the reaction is second-order, and so on. 

Graphical methods can be used to obtain the conversion from a series of reactors and have the advantage of displaying the concentration in each reactor. Moreover, no additional complications are introduced when the rate equation is not first order. As an illustration of the procedure consider three stirred-tank reactors in series, each with a different volume, operating as shown in Fig. 4-13<7. The density is constant, so that at steady state the volumetric flow rate to each reactor is the same. The flow rate and reactant concentration of the feed Q and Cq) are known, as are the volumes of each reactor. We construct a graph of the rate of reaction r vs reactant composition. The curved line in Fig. 4- 3b shows how the rate varies with C according to the rate equation, which may be of any order. 

Similar methods may be used for analysis of selectivity, yield, and concentration maxima in other types of parallel or sequential reaction schemes with either reversible or irreversible steps. However, when the kinetics involve other than first-order rate laws, convenient analytical solutions cannot be obtained, in general, and step-by-step calculations are probably more convenient. Also, the graphical method illustrated in Figure 4.15 is not applicable for parallel or sequential schemes because normally the rate of appearance or consumption of intermediates of interest depends on the concentration of more than one species and representation of rate in a single (—r) versus C relationship is not possible. 

A novel attempt for the better utilization of the normal pulse polaro-graphic method in the study of follow-up reaction kinetics has been proposed by Kim [117], This technique is based on symmetry analysis of the first and higher-order derivative NP polarographic curves. Reversible electron transfer coupled with a first-order irreversible following reaction (ECirr mechanism) was assumed. Significant effects were expected for fast chemical steps. This supposition is true for benzidine rearrangement in the course of the reduction of nitrobenzene [18, 116, 117]. 

We can apply this graphical method to the integrated rate equation to help us determine reaction order from concentration-versus-time data. Let us rearrange the integrated first-order rate equation... 

Graphical methods can also be used to determine the rate constant for zero-order reactions. Table 14.2 snmmarizes this and the other relationships discnssed in this section for zero-order, first-order, and second-order reactions. 

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