Rate equation analysis

For a sensor with good electrodes (high Jg), the response time is determined by the characteristics of the diffusion barrier and the sensor structure. For the sensor of Fig.6b, for example, a simple rate equation analysis for Pv, the oxygen pressure inside v, predicts that for a cylindrical aperture... 

Using the differential method of rate equation analysis, estimate the reaction order n and rate constant k. 

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... 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

As with the case of energy input, detergency generally reaches a plateau after a certain wash time as would be expected from a kinetic analysis. In a practical system, each of its numerous components has a different rate constant, hence its rate behavior generally does not exhibit any simple pattern. Many attempts have been made to fit soil removal (50) rates in practical systems to the usual rate equations of physical chemistry. The rate of soil removal in the Launder-Ometer could be reasonably well described by the equation of a first-order chemical reaction, ie, the rate was proportional to the amount of removable soil remaining on the fabric (51,52). In a study of soil removal rates from artificially soiled fabrics in the Terg-O-Tometer, the percent soil removal increased linearly with the log of cumulative wash time. 

FIG. 7-2 Linear analysis of catalytic rate equations, a), (h) Sucrose hydrolysis with an enzyme, r = 1curve-fitted with a fourth-degree polynomial and differentiated for r — (—dC/dt). Integrated equation,... 

Accordingly, the change in concentration (or in temperature) across the reactor can be made as small as desired by upping the recycle ratio. Eventually, the reac tor can become a differential unit with substantially constant temperature, while substantial differences will concurrently arise between the fresh feed inlet and the produc t withdrawal outlet. Such an operation is useful for obtaining experimental data for analysis of rate equations. 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

Analysis of the rate equation and kinetic model of the conversion of glucose to gluconic acid is discussed in Chapter 11. 

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 ... 

The experimental study of solid eatalyzed gaseous reaetions ean be performed in bateh, eontinuous flow stirred tank, or tubular flow reaetors. This involves a stirred tank reaetor with a reeyele system flowing through a eatalyzed bed (Figure 5-31). For integral analysis, a rate equation is seleeted for testing and the bateh reaetor performanee equation is integrated. An example is the rate on a eatalyst mass basis in Equation 5-322. 

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. 

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

The initial goal of the kinetic analysis is to express k as a function of [H ], pH-independent rate constants, and appropriate acid-base dissociation constants. Then numerical estimates of these constants are obtained. The theoretical pH-rate profile can now be calculated and compared with the experimental curve. A quantitative agreement indicates that the proposed rate equation is consistent with experiment. It is advisable to use other information (such as independently measured dissociation constants) to support the kinetic analysis. 

The rate equation (6-79), upon which this analysis has been based, may be found... 

In contrast to consecutive reactions, with parallel competitive reactions it is possible to measure not only the initial rate of isolated reactions, but also the initial rate of reactions in a coupled system. This makes it possible to obtain not only the form of the rate equations and the values of the adsorption coefficients, but also the values of the rate constants in two independent ways. For this reason, the study of mutual influencing of the reactions of this type is centered on the analysis of initial rate data of the single and coupled reactions, rather than on the confrontation of data on single reactions with intergal curves, as is usual with consecutive reactions. 

The results obtained showed, again, that the form of the rate equations and the values of their constants, obtained by the study of isolated reactions, are valid also in the coupled system. This was also confirmed by the observed agreement between the calculated and the experimental integral data (94)- Kinetic results and the analysis of the effect of reaction products revealed that adsorption of the reaction components was competitive and that all the compounds involved in the three reactions were adsorbed on the same sites of the catalytic surface. 

An analysis of the rate of release of adsorbed atoms from sites with a continuous energy spectrum for the case of an arbitrary distribution function of initial site populations was given by Carter (32). The rate equation for the t th desorption process with x = 1 and negligible readsorption is... 

It is sometimes found that a given set of a—time observations are obeyed with equal accuracy by two different rate equations and the kinetic analysis resolves itself into a test of distinguishing the applicability of the alternative functions of a. Four general approaches have been used in kinetic analyses. 

For most real systems, particularly those in solution, we must settle for less. The kinetic analysis will reveal the number of transition states. That is, from the rate equation one can count the number of elementary reactions participating in the reaction, discounting any very fast ones that may be needed for mass balance but not for the kinetic data. Each step in the reaction has its own transition state. The kinetic scheme will show whether these transition states occur in succession or in parallel and whether kinetically significant reaction intermediates arise at any stage. For a multistep process one sometimes refers to the transition state. Here the allusion is to the transition state for the rate-controlling step. 

Kinetic analysis based on the Langmuir-Hinshelwood model was performed on the assumption that ethylene and water vapor molecules were adsorbed on the same active site competitively [2]. We assumed then that overall photocatalytic decomposition rate was controlled by the surface reaction of adsorbed ethylene. Under the water vapor concentration from 10,200 to 28,300ppm, and the ethylene concentration from 30 to 100 ppm, the reaction rate equation can be represented by Eq.(l), based on the fitting procedure of 1/r vs. 1/ Ccm ... 

Sharma et al. [153] have devised a gentle accelerated corrosion test using a kinetic rate equation to establish appropriate acceleration factors due to relative humidity and thermal effects. Using an estimate for the thermal activation energy of 0.6 eV and determining the amount of adsorbed water by a BET analysis on Au, Cu and Ni, they obtain an acceleration factor of 154 at 65°C/80% RH with respect to 25 °C/35-40% RH. 

This equation is fundamental to all aspects of the kinetics of enzyme action. The Michaelis-Menten constant, KM, is defined as the concentration of the substrate at which a given enzyme yields one-half of its maximum velocity. is the maximum velocity, which is the rate approached at infinitely high substrate concentration. The Michaelis-Menten equation is the rate equation for a one-substrate enzyme-catalyzed reaction. It provides the quantitative calculation of enzyme characteristics and the analysis for a specific substrate under defined conditions of pH and temperature. KM is a direct measure of the strength of the binding between the enzyme and the substrate. For example, chymotrypsin has a Ku value of 108 mM when glycyltyrosinylglycine is used as its substrate, while the Km value is 2.5 mM when N-20 benzoyltyrosineamide is used as a substrate... 

The rate coefficients for H" in the autoionizing states (s, p) may differ from those in higher /-states and are treated differently (subscripts 0 and 1 ). A steady-state analysis of the rate equations gives a deionization coefficient of ... 

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