Differential rate equation Tables

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

The differential rate equations, corresponding integral rate equations and rate constants for various reactions (having order zero to three) under different sets of conditions are summarized in Table 1.1. 

The simplest reactions have the one-step unimolecular or bimolecular mechanisms illustrated in Table 4.1 along with their differential rate equations, i.e. the relationships between instantaneous reaction rates and concentrations of reactants. That simple unimolecular reactions are first order, and bimolecular ones second order, we take as self-evident. The integrated rate equations, which describe the concentration-time profiles for reactants, are also given in Table 4.1. In such simple reactions, the order of the reaction coincides with the molecularity and the stoichiometric coefficient. 

Onken and Matheson (1982) studied kinetics of phosphorus dissolution in EDTA (ethylenediamine tetraacetic acid) solution for several soils. They examined eight kinetic models (Table 2.2) and found that phosphorus dissolution in EDTA solution was best described using the two-constant rate, Elovich, and differential rate equations as indicated by high r2 and low SE values. None of the models best described the dissolution for all soils. 

For this example and seven others from this book, Table 11.1 illustrates the reduction of complexity achieved, showing a comparison of the numbers of rate and other equations and their coefficients of reduced and "brute force" models. The latter are understood to consist of the rate equations for all participants except those that can be replaced by stoichiometric constraints, and the constraints used in this fashion. The greatest reductions are where it counts most in the possibly differential rate equations. Also important is the reduction in the number of coefficients. This is because the problem with brute-force modeling today is not so much the demands of the actual calculations, but the experimental work required to obtain values for all the coefficients and their activation energies. 

Slightly more complex are constraints with respect to the feasible intervals that are induced by interactions between metabolites. Until now, all saturation parameters were chosen independently, using a uniform distribution on a given interval We emphasize that this choice indeed samples the comprehensive parameter spaces, and for all samples, there exists a system of explicit differential equations that are consistent with the sampled Jacobian. However, obviously, not all rate equations can reproduce all sampled values. In particular, competition between substrates for a single binding site will prohibit certain combinations of saturation values to occur. For example, consider an irreversible monosubstrate reaction with competitive inhibition (see Table II) ... 

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. 

Table 3.1 Differential and integrated rate equations for irreversible processes. ...

The results at other conversions are shown in the third column of Table 4-4. Although there is some variation from point to point, there is no significant trend. Hence the differential method also confirms the validity of a second-order rate equation. The variation is due to errors associated with the measurement of slopes of the curve in Fig. 4-1. 

The data in Table 2.5 were obtained in a differential reactor at 150°C for the reaction A + B C. Obtain an empirical rate equation and discuss possible controlling mechanisms. 

In order to provide data for testing the method, the rate equation in differential form (Eq. (8.8)) was solved numerically by means of a fourth-order Runge—Kutta method for specific values of n, E, and A. In the calculations, it was assumed that E = lOOkj/mol and A/fi = 3 X 10 min . The equation was solved with n assumed to have a value of 5/3, and the calculated values for Ea/R are shown in Table 8.2. 

TABLE 11.5 Cleland nomenclature for bisubstrate reactions exemplified. Three common kinetic mechanisms for bisubstrate enzymatic reactions are exemplified. The forward rate equations for the order bi bi and ping pong bi hi are derived according to the steady-state assumption, whereas that of the random bi bi is based on the quasi-equilibrium assumption. These rate equations are first order in both A and B, and their double reciprocal plots (1A versus 1/A or 1/B) are linear. They are convergent for the order bi bi and random bi bi but parallel for the ping pong bi bi due to the absence of the constant term (KiaKb) in the denominator. These three kinetic mechanisms can be further differentiated by their product inhibition patterns (Cleland, 1963b)... 

We have now determined the reaction order of each reactant for ethyl acetate saponification via two procedures one, with respect to initial reactant concentration the other, with respect to initial reaction time. We have a choice with regard to the latter procedure we can calculate the reaction rate using a central difference algorithm or we can generate a polynomial equation describing the change in reactant concentration as a function of time, then differentiate that polynomial to obtain the reaction rate. Table 2.4 presents the results for each of these procedures for ethyl acetate saponification. So— which reaction rate equation is correct The more appropriate question is which reaction rate equation is valid The most valid reaction rate equation is the one based on initial reactant concentrations. The reaction rate equations based on initial reaction time are approximations of it. The sets of equations will be similar but not necessarily the same since we determined each set via a different experimental procedure. 

The experimental rate law can be determined by monitoring the concentration of one of the reactants or products as a function of time using spectroscopic means. For instance, the Beer-Lambert law states that the absorbance of a colored compound is directly proportional to its concentration (for optically dilute solutions anyway), so that the absorbance can be measured as the course of the reaction proceeds. The data are then fit to a model, such as the function that results when integrating one of the differential rate law equations. The integrated rate laws for some commonly occurring kinetics are listed in Table 17.1. Half-life equations are also included for some of the reactions in this table, where the half-life ftyi) is defined as the length of time that it takes for half of the initial reactant concentration to disappear. 

O2 (760 Torr = 1 atm) under differential reactor conditions, an apparent activation energy of 29 kcal mole for CO2 formation was observed. Near 900 K, selectivity to CO2, rather than CO, was about 75% or higher, and the reaction orders from a power rate law are given in Table 1. Propose a L-H-type model for CO2 formation with a sequence of elementary steps that results in a derived rate expression consistent with these results. It can be assumed that only the adsorbed reactants and products need be included in the site balance, and dissociative O2 adsorption occurs. Under low-conversion conditions, the surface concentrations of the products can be ignored, so what is the form of the rate equation Fitting this latter equation to the data produced the optimized rate parameters listed in Table 2, where k is the lumped apparent rate constant. Evaluate them to determine if they are consistent and state why. 

Figure 6-1 compares the rates obtained with the two methods of differentiation discussed above. The data in Table 6-2 were generated using the second-order rate equation ... 

Figure 6-1 Comparison of the values of the reaction rate —ta obtained by different methods of differentiation. The unfilled points are the results from numerical differentiation (Procedure 1). The filled points are the results from fitting a polynomial to the concentration versus time data in Table 6-2 and differentiating the polynomial analytically (Procedure 2). The solid line is the second-ordo rate equation —ta = 0.2368 xCl, which was used to generate the data in Table 6-2.

---