The differential method

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 ... 

If the reaetion rate depends on more than one speeies, use the method of exeess eoupled either with the half-life method or the differential method. If the method of exeess is not suitable, an initial rate plot may be eonstrueted by varying the eoneentration of one reaetant while the eoneentrations of the others are held eonstant. This proeess is repeated until the orders of reaetion of eaeh speeies and the speeifie reaetion rate are evaluated. At level 5, the least-squares analysis ean be employed. 

Prepare a plot of reaction rate (-dC /dt) versus f(C ). If the plot is linear and passes through the origin, the rate equation is consistent with the data, otherwise another equation should be tested. Figure 3-17 shows a schematic of the differential method. 

Figure 3-17. Sohematios of the differential method for data analysis.

After the rates have been determined at a series of reactant concentrations, the differential method of testing rate equations is applied. Smith [3] and Carberry [4] have adequately reviewed the designs of heterogeneous catalytic reactors. The following examples review design problems in a plug flow reactor with a homogeneous phase. 

For many situations, a simple total anthocyanin determination is inappropriate because of interference from polymeric anthocyanins, anthocyanin degradation products, or melanoidins from browning reactions. In those cases, the approach has been to measure the absorbance at two different pH values. The differential method measures the absorbance at two pH valnes and rehes on structural transformations of the anthocyanin chromophore as a function of pH. Anthocyanins switch from a saturated bright red-bluish color at pH 1 to colorless at pH 4.5. Conversely, polymeric anthocyanins and others retain their color at pH 4.5. Thus, measurement of anthocyanin samples at pH 1 and 4.5 can remove the interference of other materials that may show absorbance at the A is-max-... 

In the above-described measurement, which we call the absolute method, all pumps have equal speeds (rpm) owing to interconnection to the same drive-shaft. In order to express, if required, a deviation registered for the analyte concentration, one must calibrate with a standard by varying its rpm (B) with respect to that of the titrant (A) a B/A rpm ratio greater than unity means a proportionally lower concentration and vice versa. In general, the absolute method serves to control a sample stream with nearly constant analyte concentration as a sensor one uses not only electroanalytical but often also optical detectors. However, with considerably varying analyte concentrations the differential method is more attractive its principle is that in the set-up in Fig. 5.15 and with the sensor adjusted to a fixed and most sensitive set-point, the rpm of the sample stream (C) is varied with respect to that of the titrant (A) by a feedback control (see Fig. 5.3a) from the sensor via a regulator towards the... 

The following example illustrates the use of the differential method for the analysis of kinetic data. It also exemplifies some of the problems... 

Of the methods available for potentiometrically estimating the amount of iodine bound by amylose, the differential method of Gilbert and Marriott36 is by far the most satisfactory for accurate work as it eliminates corrections for reagent blanks. In this method, the amylose solution and control solution form two half-cells connected by a salt-bridge, and the... 

To determine the form of the rate law, values of (-rA) as a function of cA may be obtained from a series of such experiments operated at various conditions. For a given reactor (V) operated at a given % conditions are changed by varying either cAo or q. For a rate law given by (—rA) = kAcA, the parameter-estimation procedure is die same as that in the differential method for a BR in the use of equation 3.4-2 (linearized form of the rate law) to determine kA and n. The use of a CSTR generates point ( -rA) data directly without the need to differentiate cA data (unlike the differential method with aBR). 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

An additional 3 % of the linters was also estimated to be accessible. This additional material, which hydrolyzed between. 07 and 1 hour, was regarded as fairly well ordered since its removal was effected less readily than the first 3 % and since it exhibited a hygroscopic behavior similar to highly acid-resistant cellulose. By difference then, the crystalline material should constitute 94% of the intact linters. A value of 92% crystalline or highly resistant cellulose was derived from the weights of residues recovered. This result confirmed, at least qualitatively, the amounts found by the differential method. 

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... 

In the differential method of analysis we test the fit of the rate expression to the data directly and without any integration. However, since the rate expression is a differential equation, we must first find llV) dNldt) from the data before attempting the fitting procedure. 

There are advantages and disadvantages to each method. The integral method is easy to use and is recommended when testing specific mechanisms, or relatively simple rate expressions, or when the data are so scattered that we cannot reliably find the derivatives needed in the differential method. The differential method is useful in more complicated situations but requires more accurate or larger amounts of data. The integral method can only test this or that particular mechanism or rate form the differential method can be used to develop or build up a rate equation to fit the data. 

In general, it is suggested that integral analysis be attempted first, and, if not successful, that the differential method be tried. 

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. 

The differential method of analysis deals directly with the differential rate equation to be tested, evaluating all terms in the equation including the derivative dCJdt, and testing the goodness of fit of the equation with experiment. 

Figure 3,17 Test for the particular rate form —r = kf Cp by the differential method.

Figure 3.18 Test for an th-order rate form by the differential method.

Repeat the above problem, except this time solve by the differential method. 

Differential Analysis. Integral analysis provides a straightforward rapid procedure for testing some of the simpler rate expressions. However, the integrated forms of these expressions become unwieldy with more complicated rate expressions. In these situations, the differential method of analysis becomes more convenient. The procedure is closely analogous to the differential method described in Chapter 3. So, by differentiating Eq. 53 we obtain... 

The differential method was also applied for processing the experimental data. The result was in good agreement. 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

In the differentiation method, values of the instantaneous reaction rate per unit volume (l/y)(dA/,/dt) are obtained directly from experimental data points by differentiation and fitted to an assumed rate equation. 

Each of these methods has both merits and demerits. For example, the integration method can easily be used to test several well-known specific mechanisms. In more complicated cases the differential method may be useful, but this requires more data points. Analysis by the integration method can generally be summarized as follows. 

The differential method (see Basic Protocol 1) measures the absorbance at two different pH values, and relies on the structural transformations of the anthocyanin chromophore as a function of pH (Fig. Fl.2.1 and Fig. FI.2.2). This concept was first introduced by Sondhe-imer and Kertesz in 1948, who used pH values of 2.0 and 3.4 for analyses of strawberry jams (Francis, 1989). Since then, the use of other pH values has been proposed. Fuleki and Francis (1968b) used pH 1.0 and 4.5 buffers to measure anthocyanin content in cranberries, and modifications of this technique have been applied to a wide range of commodities (Wrolstad et al., 1982, 1995). The pH differential method has been described as fast and easy for the quantitation of monomeric anthocyanins (Wrolstad et al., 1995). 

Gordon Campbell (Ref 17) described the differential method of analysis in the investigation of thermal decompn of inorganic oxidants, such as AN... 

Fig. 3.1 Illustration of how to use the differential method (a) for a reagent disappearing (b) for a product appearing (c) [reactant] versus f for different initial concentrations showing the slopes at f = 0 (d) plot of the logarithms of the different slopes obtained in (c) versus logarithms of the initial concentrations, i.e. In ro,/, versus Inpeactantlo, .

Determination of the rate constant by the differential method is straightforward either of the above-mentioned plots used to determine a and ft gives k from the intercept. 

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