Representative initial rate data

Table 1. Representative initial rate data used to determine the order of reaction for the reactants in eq. 3.

A La(Cr, Ni) 0, catalyst was tested for the cleanup of residual hydrocarbons in combustion streams by measuring the rate of methane oxidation in a differential laboratory flow reactor containing a sample of the catalyst. The following conversions were measured as a function of temperature with a fixed initial molar flow rate of methane. The inlet pressure was 1 bar and the methane mole fraction was 0.25. (Note that the conversions are small, so that the data approximately represent initial rates.) The rate law for methane oxidation is first-order with respect to methane concentration. 

In cases where sorptive equilibrium is reached rapidly and transformation is much slower, the aqueous phase concentration of contaminant may show a rapid initial decrease due to adsorption followed by a slower decline due to transformation. Under these conditions, the kinetic model represented by Eq. (16) is sufficient to describe the kinetics of transformation after the initial data have been excluded. This approach has been taken for TCE [168], vinyl chloride [176], and probably in many other studies where the exclusion of initial rate data was not clearly documented. 

Tlie usual procedure for the graphical analysis of initial rate data would be to treat each substrate as the varied substrate at different fixed concentrations of another substrate, maintaining a fixed concentration of the third substrate. All such plots represent a family of straight lines with a common intersection point to the left of the i/Uo-axis. 

A reaction scheme has been proposed for the catalysis of the disproportionation of H2O2 by the cobalt(ra)-haematoporphyrin complex of haemato-porphyrin(IX), [Co HP], which represents a model for catalase-like activity. Unlike other cobalt(m) complexes, the axial co-ordinating positions are labile in this complex. In the pH range 5.5—6.5, initial rate data exhibit an inverse dependence on hydrogen-ion concentration, the rate data being consistent with the reactions... 

Figure 8 Plot of the initial rate of the enzyme-catalyzed oxidation of 1-phenylpropanol as a function of % ee. The solid line represents a fit of the data to the Michaelis-Menten formalism for competitive inhibition where [S] = [ -(60)] and [ ] = [ -(60)]. The total alcohol concentration was maintained constant at lOmM.100...

By using only simple hand calculations, the single-site model has been rejected and the dual-site model has been shown to represent adequately both the initial-rate and the high-conversion data. No replicate runs were available to allow a lack-of-fit test. In fact this entire analysis has been conducted using only 18 conversion-space-time points. Additional discussion of the method and parameter estimates for the proposed dual-site model are presented elsewhere (K5). Note that we have obtained the same result as available through the use of nonintrinsic parameters. 

In every case in which a kinetic model is selected to represent adequately a reaction, the rate surface predicted by the model must be compared to the surface observed in the data. In the methods discussed in Section II, only one section through the entire rate surface was examined for example, the dependence of initial rate on total pressure could be investigated when in fact the total rate surface constituted the dependence of rate on several component partial pressures and temperature. The misleading results obtain-... 

The effects of various formulation factors on the in vitro release characteristics of spherical polymethylmethacrylate implants were studied. Physical and mathematical models were proposed to describe the in vitro release profiles. The in vitro release data could be described by a biexponential equation of the following type fraction of tobramycin remaining in the implant at time t=Aerai+BQ, where a, and P represent the rate constants for the initial rapid and subsequent slow phases of release. The influence of drug loading, volume of dissolution medium, implant size and type of cement and the incorporation of water-soluble additives on the release profiles and a and P rate constants is described. 

The existing data show that the rate of cationic homopolymerization is increased in most cases by the application of an electric field. Fig. 1 shows typical results, in which RPE and Rpo are the initial rates of polymerization Rp) with and without electric field, respectively (5). Figs. 2 and 3 represent the dependences of Rp on monomer and catalyst concentrations, respectively (9,10). It is evident that Rt is increased under an electric field, whereas the concentration dependences (the reaction orders) are not influenced. It is likely therefore that the rate enhancement is not due to a new reaction mechnism but rather to an increase... 

For SK-500 the rate at 573°K and 400 sec after the initiation of reactant flow is independent of reactant mole ratio for Ce C2 = 0.7 to 10. Under these conditions the 400-sec point is just beyond the maximum in the rate curve. Similar behavior was observed at one other condition. Initial rate of reaction estimated by extrapolating the decay portion of the rate curves for this data to zero time (see below) indicates a maximum in the rate at C6 C2 == 3.5 (Figure 2). Error bars represent estimated 95% confidence limits. The observed activity for HY is about twice that of SK-500, that for LaY is about two-thirds that of SK-500 (Figure 2). This is consistent with the trend expected (7) since all catalysts were activated to the same temperature. The temperature dependence of the observed rate is large for all systems studied indicating the absence of external mass transfer limitations. 

Fig. 4-2. Effect of NO concentration, N02 partial pressure, and temperature on reaction rate of NO oxidation, with oxygen partial pressure between 6 and 8 torr. Tie lines indicate slight effect of N02 on initial rate. Dashed curves have slope two and are shown for reference purposes. Both axes are on a logarithmic scale. Lower curves on plots for 2 and 4 torr N02 represent initial (time zero) data. Tie lines connect points where all other variables except [NOa] are constant (from Treacy and Daniels429 with permission of the American Chemical Society).

The effective local temperatures in both sites were determined. By combining the relative sonochemical reaction rates for equation 5 with the known temperature behavior of these reactions, the conditions present during cavity collapse could then be calculated. The effective temperature of these hotspots was measured at 5200 K in the gas-phase reaction zone and 1900 K in the initially liquid zone (6). Of course, the comparative rate data represent only a composite temperature during the collapse, the temperature has a highly dynamic profile, as well as a spatial temperature gradient. This two-site model has been confirmed with other reactions (27,28) and alternative measurements of local temperatures by sonoluminescence are consistent (7), as discussed later. 

Figure 1 Time- and concentration-dependent inactivation of the catalytic activity of P450 2B6 by bergamottin. Inactivation of the EFC O-deethylation activity of P450 2B6 in the reconstituted system incubated with 0.6 ( ), 1 (o), 2 ( ), 3 (O), 5 ( ), and 10 ( ) pM bergamottin. Aliquots were removed at the indicated time and assayed for residual activity. The insets show the double reciprocal plots of the initial rates of inactivation as a function of the bergamottin concentrations. The kinetic constants Kh inact, and f1/2 were determined from this plot. The data shown represent the average of three experiments that did not differ by more than 10%. Source From Ref. 72.

Because of the extreme dependence on initial conditions, our history analysis concentrates on an air mass with relatively well-defined concentrations at the beginning and the end of its travel. Giving it the initial values, we see the concentrations unfold as the air parcel moves through the computed simulation procedure. Because of the sensitivities discovered, the transition of oxidant species O3 and NO2 proceeds better than one might expect. The previously adopted biases on the NO-flux and the propylene oxidation rates were confirmed in this run having different conditions from those in Huntington Park represented by 1968 data. 

Figure 2 The relationship between /, the chondrite normalized Stn/ Nd, eNj evolution, and time. The curves represent the rate of change of SNd for different / values in the Early Archean mantle. Atime is the difference between the age of the Earth ( 4.5 Ga) and the sample age. The shaded area shows the range of initial cnj compositions that have been proposed for 3.8 Ga mantle based on data from Early Archean gneisses. Depending on the age of the mantle fractionation event, the Early Archean upper mantle may have been equally, or even more TREE depleted than the modern N-MORB source mantle.

However, it is often the case that not all the concentrations which are required for the calculation of the reaction stoichiometry are available. It may be that experimentally it is only practicable to measure the concentration of, say, the reactant A at a series of times subsequent to the reaction being started. In this situation, our only help comes from the initial rate. As before, if the initial rate of disappearance of A is affected by the presence of product and/or is less than the rate observed somewhat later in the reaction, we know immediately that the more complicated expressions are required. The converse observations, however, do not necessarily imply that the rate can be represented by the simple equation, (1). In those cases where the preliminary examination of the data indicates that the maximum rate occurs at zero time and is unaffected by the presence of product, the data are examined on the basis of eqn. (1) first if inconsistent rate coefficients are found together with non-simple orders, the data are then re-examined on the basis of the more complicated expressions. This same procedure is adopted when the data consist of a series of concentrations and times but where, for experimental reasons, the early concentration-time values are either unobtainable or sufficiently unreliable as to preclude any reasonable estimates of the initial rate being made the data are examined on the basis of the simple expression first and in the event of inconsistencies re-examined on the basis of the more complicated expressions. It should be clear that improvements in the experimental technique designed to reduce the uncertainties in the initial rates can more than repay the effort involved. 

The empirical rate law in Eq. 23 holds only for the initial rates. Tamura et al. (1976) observed an autocatalytic effect of the ferric precipitates produced in the reaction. Sung and Morgan (1980) identified y-FeOOH as the primary oxidation product at neutral pH and confirmed its autocatalytic effect. Adsorbed Fe(II) seems to compete in an additional parallel reaction with the dissolved ferrous species. Fast surface reaction rates resulted from a fit of the kinetic data. Examples of these constants are included in Fig. la for comparison. They represent only estimates of an order of magnitude because Tamura et al. (1976) did not determine the surface concentration of Fe(II). However, Figure 2 shows qualitatively that the ferrous ion is adsorbed specifically to mineral surfaces. 

Fig. 9 Initial rate of hydrolysis of methyl butyrate by soluble lipase closed square), lipase immobilized on green coconut fiber closed circle) and Novozyme 435 closed triangle) in fully aqueous medium. The solid line represents the fit of Michaelis-Menten model to experimental data. The dashed lines represent a linear regression of the experimental data. The amount of adsorbed protein (50 mg/g of catalyser) in Novozyme 435 was determined by Secundo et al. [42]...

If the surface is first saturated with a monolayer of protein exposed to steady-state concentration cQ, and then is exposed to a second treatment at concentration 2c0, a second front emerges. The second profile represents the situation where no net protein is adsorbed and thus, in principle, is representative of the diffusion-shifted flow pattern of the nonadsorbed protein. Figure 7 shows both the initial (cQ) and second (2c0) fronts and the subtraction curve which is very close to the ideal step function. If the data are interpreted as solution-borne molecules passing over an inert surface, then (a) adsorption must be essentially instantaneous and (b) the surface must become covered by exhausting the concentration of solute at the front as it moves down the column. The slope of the difference profile should represent the rate of uptake of material on the column, and that is essentially infinite on the time scale of the experiment. The point of inflection of the subtracted front indicates the slowing of the sorption process due to filling of sites on the surface. 

The information required here is not concentration versus time, but rate of reaction versus concentration. As will be seen later, some types of chemical reactors give this information directly, but the constant-volume, batch systems discussed here do not [ What does it profit you, anyway —F. Villon], In this case it is necessary to determine rates from conversion-time data by graphical or numerical methods, as indicated for the case of initial rates in Figure 1.25. In Figure 1.27 a curve is shown representing the concentration of a reactant A as a function of time, and we identify the two points Cai and Ca2 for the concentration at times q and t2- The mean value for the rate of reaction we can approximate algebraically by... 

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