Integrated rate methods

The data is concentration/time data and is immediately adaptable to integrated rate methods. To use the methods of Section 3.7 the data has to be rate/ concentration data, which is not the manner in which it is presented in this problem. 

Neither (a) the differential method, based on rate/concentration data, nor (b) the integrated rate method, based on concentration/time data, is easily applicable. 

The integrated rate method has also been shown (Section 3.11.2) to become cumbersome to use if the reactant concentrations are different. 

An alternative treatment nses the integrated law, eq. (4.12), to study the kinetics of this reaction. The graph of the reciprocal of concentration as a fnnction of time gives a straight hne whose slope is 2 (Figure 4.2). Again, the integrated rate method is qnicker and less snbjective than the differential form. 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Sensitivity The sensitivity for a one-point fixed-time integral method of analysis is improved by making measurements under conditions in which the concentration of the monitored species is larger rather than smaller. When the analyte s concentration, or the concentration of any other reactant, is monitored, measurements are best made early in the reaction before its concentration has substantially decreased. On the other hand, when a product is used to monitor the reaction, measurements are more appropriately made at longer times. For a two-point fixed-time integral method, sensitivity is improved by increasing the difference between times t and f2. As discussed earlier, the sensitivity of a rate method improves when using the initial rate. 

A reading of Section 2.2 shows that all of the methods for determining reaction order can lead also to estimates of the rate constant, and very commonly the order and rate constant are determined concurrently. However, the integrated rate equations are the most widely used means for rate constant determination. These equations can be solved analytically, graphically, or by least-squares regression analysis. 

The most widely used method for fitting a straight line to integrated rate equations is by linear least-squares regression. These equations have only two variables, namely, a concentration or concentration ratio and a time, but we will develop a more general relationship for use later in the book. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

Numerical approaches for estimating reactivity ratios by solution of the integrated rate equation have been described.124 126 Potential difficulties associated with the application of these methods based on the integrated form of the Mayo-kewis equation have been discussed.124 127 One is that the expressions become undefined under certain conditions, for example, when rAo or rQA is close to unity or when the composition is close to the azeotropic composition. A further complication is that reactivity ratios may vary with conversion due to changes in the reaction medium. 

Wilkinson s method allows the evaluation of the reaction order from data taken during the first half-life. This, as we saw, was not possible from treatment by the integrated rate law. Note, however, that relatively small errors in [A] can lead to a larger error in E at small conversions.17... 

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. 

Integral control describes a controller in which the output rate of change is dependent on the magnitude of the input. Specifically, a smaller amplitude input causes a slower rate of change of the output. This controller is called an integral controller because it approximates the mathematical function of integration. The integral control method is also known as reset control. 

In this study five cellulose samples of different crystallinities (10, 41, 63, 67, and 742) were treated to 10% by weight with H PO, H3BO3, and AlClo i O. These treated samples and untreated (control) samples were isothermally pyrolyzed under N2 at selected temperatures and the TGA data analyzed by four methods (0—, 1st-, and 2nd-order and Wilkinson s approximation) to obtain rates of mass loss. From these rates, activation energy (Efl), activation entropy (AS+) and enthalpy (AH+) values were obtained. Efl was also determined by the integral conversion method. 

Ea s were also determined by the integral conversion method (17). This method does not require assumption of order or determination of rate constants. The integral conversion method may have limited usefulness since the values obtained did not always agree with the Efl values obtained by the Arrhenius equation of the 0—, 1st- or 2nd-order constants. 

Trial and error method A rate equation which describes the experimental points with the best fit is chosen. The differential and integrated rate equations for the various reaction orders are found in Table 4-2. The best fit is easy to find by comparing the linear regression coefficients for the appropriate x y-pairs. The x-axis is always the time t. 

Another group of methods for the analysis of data from temperature scans at constant heating rate is based on the integrated rate Eq. 

Experimentally observed quantities pertaining to the whole surface, such as the amount of adsorbed substance, heat of adsorption, reaction rate, are sums of contributions of surface sites or, since the number of sites is extremely great, the respective integrals. As increases monotonously with s, each of them can be taken as variable for integration both methods of calculation are used. If is chosen as an independent variable, a differential function of distribution of surface sites with respect to desorption exponents, [Pg.211]

Improvements in membrane technology, validation of membrane integrity, and methods to extend filter usage should further improve the performance of membrane filters in removal of viral particles. Methods to improve or extend filter life and increase flow rates by creating more complex flow patterns could possibly be the focus of the next generation of membrane filters designed to remove viral particles. 

Rate equations like 2.27 and 2.28, obtained from a proposed set of elementary reaction steps, are differential equations. Although for our purposes in this book we shall require only differential rate equations, it is usually more convenient in interpreting raw experimental data to have the equations in integrated form. Methods of integration of rate equations can be found in the literature.34... 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

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