Initial rate curves

Figure 8. Arrhenius diagram of the gamma-ray initiated polymerization of acrylonitrile in bulk 1161 Curve 1 Initial rates" curve 2 Pseudostationary rates. ...

Figure 2. The dependence of AG polymerization initial rate (curve 1) ([APS] = 5x103 mole l 1 60°C) and intrinsic viscosity of resulted polymers (curve 2) (LVNaCl solution in ff20 30°C) on monomer concentration.

This method makes use of comparison between the theoretical and the experimental initial rate curves. Rate equations derived from the mechanisms described above were expressed as a function of total pressure (Pt) and conversion (X). Then the substitution of X= gave the parametric form of the initial rate (lO) which is plotted against total pressure in Figure 1. 

The experimental initial rate curves in the Berty reactor (Fig. 2.b) shows a trend that can represent either type b ox e among the theoretical graphs in the Figure 1. Since the model Ml 7 belongs to type c it can be removed in the final selection from likely mechanisms. The two retained models. Ml 8 and M20 both are of Mars-van Krevelen types where the steady-state assumption involves the equality of the rate of three elementary steps as it has been shown in the Table 1. 

A considerable assumption in the exponential distribution is the assumption of a constant failure rate. Real devices demonstrate a failure rate curve more like that shown in Eigure 9. Eor a new device, the failure rate is initially high owing to manufacturing defects, material defects, etc. This period is called infant mortaUty. EoUowing this is a period of relatively constant failure rate. This is the period during which the exponential distribution is most apphcable. EinaHy, as the device ages, the failure rate eventually increases. 

Casado et al. have analyzed the error of estimating the initial rate from a tangent to the concentration-time curve at t = 0 and conclude that the error is unimportant if the extent of reaction is less than 5%. Chandler et al. ° fit the kinetic data to a polynomial in time to obtain initial rate estimates. 

However, as the pH—rate plot shows, at very low pH the observed rate actually decreases. Because, as the preceding argument shows, rate-determining dehydration should result in a pH-dependent rate at low pH, this decreased rate must mean that the rds has changed. This is reasonable, for at pH values well below the pKg of hydroxylamine, the decreasing proportion of the hydroxylamine in the unprotonated form will decrease the rate of the initial addition. At some pH, then, the rate of the addition step will fall below that of the dehydration step, and the observed rate curve will lie lower than the rate predicted for the dehydration. 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

Fig. 1.85 Oxidation of high-purity iron in oxygen at differing pressures. All figures on curves are in N m . At 1-3 x 10 tol-3 x 10 N/m torr the rate is controlled by the impact of molecular oxygen at I - 3 x 10 N/m torr the initial rate of oxidation is sufficiently high to give void precipitation and the rate decreases with pressure increasing to l-3N/m at pressures greater than this the crack-heal mechanism becomes operative and the rate again increases with pressure (after Hussey and Cohen...

The inhibition analyses were examined differently for free lipase in a batch and immobilised lipase in membrane reactor system. Figure 5.14 shows the kinetics plot for substrate inhibition of the free lipase in the batch system, where [5] is the concentration of (S)-ibuprofen ester in isooctane, and v0 is the initial reaction rate for (S)-ester conversion. The data for immobilised lipase are shown in Figure 5.15 that is, the kinetics plot for substrate inhibition for immobilised lipase in the EMR system. The Hanes-Woolf plots in both systems show similar trends for substrate inhibition. The graphical presentation of rate curves for immobilised lipase shows higher values compared with free enzymes. The value for the... 

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. 

Fig. 5. Effect of the initial populations hm on a normalized desorption rate curve [criterion of resolution employed after Carter (32)]. Hyperbolic heating schedule, 0 = 1.073 X 10-< deg-1 sec-1 Ea = 40 kcal mole-1 x = 1 Tm = 670°K A7 m = 15°K. Curves 1, 2, 3, correspond to naa/nm = 1.00, 1.54, 1.85, respectively.

FIGURE 13.6 The definition of the initial rate of reaction. The orange curves show how the concentration of N20, changes with time for five different initial concentrations. The initial rate of consumption of N20-, can be determined by drawing a tangent (black line) to each curve at the start of the reaction. 

FIGURE 13.14 The characteristic shapes of the time dependence of the concentration of a reactant during a second-order reaction. The larger the rate constant, k, the greater is the dependence of the rate on the concentration of the reactant. The lower gray lines are the curves for first-order reactions with the same initial rates as for the corresponding second-order reactions. Note how the concentrations for second-order reactions fall away much less rapidly at longer times than those for first-order reactions do. 

The initial rate of polymerization was determined from the initial slopes of time-conversion curves (Fig. 1) using f-BuX/Me3 Al/MeCl systems at -40 °C. This... 

The more usual pattern found experimentally is that shown by B, which is called a sigmoid curve. Here the graph is indicative of a slow initial rate of kill, followed by a faster, approximately linear rate of kill where there is some adherence to first-order reaction kinetics this is followed again by a slower rate of kill. This behaviour is compatible with the idea of a population of bacteria which contains a portion of susceptible members which die quite rapidly, an aliquot of average resistance, and a residue of more resistant members which die at a slower rate. When high concentrations of disinfectant are used, i.e. when the rate of death is rapid, a curve ofthe type shown by C is obtained here the bacteria are dying more quickly than predicted by first-order kinetics and the rate constant diminishes in value continuously during the disinfection process. 

A, obtained if the disinfection process obeyed the first-order kinetic law. B, sigmoid curve. This shows a slow initial rate of kill, a steady rate and finally a slower rate of kill. This is the form of curve most usually encountered. C, obtained if bacteria are dying more quickly than first-order kinetics would predict. The constant, K, diminishes in value continuously during the process. 

Fig. 39.17. Schematic illustration of Michaelis-Menten kinetics in the absence of an inhibitor (solid line) and in the presence of a competitive inhibitor (dashed line), (a) Plot of initial rate (or velocity) V against amount (or concentration) of substrate X. Note that the two curves tend to the same horizontal asymptote for large values of X. (b) Lineweaver-Burk linearized plot of 1/V against l/X. Note that the two lines intersect at a common intercept on the vertical axis.

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

Referring to Fig. 1.4, the solution begins with the initial concentration conditions Aq, Bq, Cq and Dq, defined at time t = 0. Knowing the magnitudes of the kinetic rate constants k], k2, k3 and k4, thus enables the initial rates of change dCA/dt, dCfi/dt, dCc/dt and dCo/dt, to be determined. Extrapolating these rates over a short period of time At, from the initial conditions, Aq, Bq, Cq and Do, enables new values for A, B, C and D to be estimated at the new time, t = t -I- At. If the incremental time step At is sufficiently small, it is assumed that the error in the new estimated values of the concentration. A, B, C and D, will also be small. This procedure is then repeated for further small increments of time until the entire concentration versus time curves have been determined. 

The shape of the recovery curve suggests that there may be a faster initial rate of recovery of serotonin uptake sites occurring between 18 hours and 4 weeks, which is followed by a slower rate of recovery between 4 weeks and 12 months. These data indicate that more than 6 months are required for a complete recovery of serotonin uptake sites to control levels. 

Fig.4.11. The initial rates v of increase and decrease of electric conductivity of the sensor (/) and adsorbent (2), as functions of the adsorbent temperature ( z), and the Arrhenius plot (b) corresponding to curve (a). The sensor (ZnO) is kept at room temperature.

Figure 8.9 Reaction progress curve in the presence of a mechanism-based inactivator when a second aliquot of enzyme is added to the reaction solution. The reaction is allowed to reach a plateau before a second, equal concentration aliquot of enzyme is added at the indicated time point. Note that the rate of inactivation for this second aliquot of enzyme is the same as that seen in the initial progress curve.

Plots of the concentration of carboxylate formed vs. time were drawn for each copolymer, and the initial rates of hydrolysis were determined by measurement of the slope of the tangent to the curve at zero time. The pseudo-unimolecular rate constant (K) is given by ... 

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