Kinetic rate equations, sigmoid

The kinetic analysis of the sigmoid pH-rate profile will yield numerical estimates of the pH-independent parameters K, k, and k". With these estimates the apparent constant k is calculated using the theoretical equation over the pH range that was explored experimentally. Quantitative agreement between the calculated line and the experimental points indicates that the model is a good one. A further easy, and very pertinent, test is a comparison of the kinetically determined value with the value obtained by conventional methods under the same conditions. 

The second-order dependence of the TNP hydrolysis rate catalyzed by 11 fits the kinetic equation 7. A plot of the observed rate constants kobsi as a function of the pH using 0.1 mM 11 and 10 pAf TNP gave a similar sigmoidal curve to that found in the earlier NA hydrolysis with an inflection point at pH 8.3 (see Figure 10). Therefore, the same species lib is concluded to react with NA and TNP. The second-order rate constant TNP (see Eq. [8]) was extremely large 1.1 X 103 AT1 sec"1 at 25°C and pH 10.2 with I = 0.10 (NaN03). 

The data under discussion were obtained in the course of a study of the anionic polymerization of isoprene in benzene at 35.1° C. Earlier work on the polymerization using -butyl lithium as initiator in a number of hydrocarbon solvents had shown that the kinetic behaviour was complicated. For example, the isoprene concentration-time curves were sigmoidal and the ratio of the number of moles of isoprene consumed to the number of moles of n-butyl lithium reacted increased as the reaction proceeded. These and other complications showed that the rate of the reaction could not possibly be represented by an equation of the type... 

Galwey and Hood [160] showed that NajCOj.l.SHjOj decomposed in vacuum (360 to 410 K) to produce Na COj + l.SHjO + O.TSOj. ar-time curves were sigmoidal and the kinetics could be described by the Avrami-Erofeev equation with = 2 or 3. The activation energy was 112 8 kJ mol. The reaction rate between 313 and 343 K was significantly increased by the presence of small amounts of liquid water. This deceleratory reaction was fitted by the first-order equation (E, = 80 10 kJ mol ) and it was concluded that breakdown of hydrogen peroxide proceeded in the liquid water, possibly with trace amounts of impurity transition-metal ions acting as catalysts. 

Silver oxide An early (1905) study by Lewis [34] of the kinetics of decomposition of Ag20 was a notable contribution. The dissociation in oxygen (760 Torr, 593 to 623 K) showed a long induction period followed by a sigmoid nr-time curve which fitted the Prout-Tompkins equation with = 133 kJ mol. Benton and Drake [35] studied the kinetics of the reversible dissociation using a sample of finely-divided active metal. The rate of reaction at 433 K fitted the expression ... 

The r-time curves for the decomposition of anhydrous cobalt oxalate (570 to 590 K) were [59] sigmoid, following an initial deceleratory process to a about 0.02. The kinetic behaviour was, however, influenced by the temperature of dehydration. For salt pretreated at 420 K, the exponential acceleratory process extended to flr= 0.5 and was followed by an approximately constant reaction rate to a = 0.92, the slope of which was almost independent of temperature. In contrast, the decomposition of salt previously dehydrated at 470 K was best described by the Prout-Tompkins equation (0.24 < a< 0.97) with 7 = 165 kJ mol . This difference in behaviour was attributed to differences in reactant texture. Decomposition of the highly porous material obtained from low temperature dehydration was believed to proceed outwards from internal pores, and inwards from external surfaces in a region of highly strained lattice. This geometry results in zero-order kinetic behaviour. Dehydration at 470 K, however, yielded non-porous material in which the strain had been relieved and the decomposition behaviour was broadly comparable with that of the nickel salt. Kadlec and Danes [55] also obtained sigmoid ar-time curves which fitted the Avrami-Erofeev equation with n = 2.4 and = 184 kJ mol" . The kinetic behaviour of cobalt oxalate [60] may be influenced by the disposition of the sample in the reaction vessel. 

Kinetic runs in step b in Fig. 8c started with a very fast reduction of approximately e per molecule, after which a slow reductioh took place, yielding sigmoidal reduction curves. This, indicates that reduction of Co2+ to Co° is controlled by the formation and slow growth of reduction nuclei of metallic cobalt on. the surface of the reduced phase in step a (nucleation model). Initially, the reduction rate increases because of the growth of nuclei already formed and the appearance of new ones. At a certain point the reduction nuclei start to overlap at the inflection point, the interface of. the oxidized and reduced phases and the reduction rate both begin to decrease. Reduction of this type is described by the Avrami-Erofeev equation (118)... 

Fig. 9.3. A comparison between hexokinase I and glucokinase. The initial velocity (vj) as a fraction of is graphed as a function of glucose concentration. The plot for glucokinase (heavy blue line) is slightly sigmoidal (S-shaped), possibly because the rate of an intermediate step in the reaction is so slow that the enzyme does not follow Michaelis-Menten kinetics. The dashed blue line has been derived from the Michaelis-Menten equation fitted to the data for concentrations of glucose above 5 mM. For S-shaped curves, the concentration of substrate required to reach half or half-saturation, is sometimes called the Sq 5 or Kq 5, rather than K. At = 0.5, the is 5 mM, and...

For an enzyme that follows MichaeHs—Menten kinetics, R = SI. For a regulatory enzyme that gives a sigmoidal rate plot, Rj < 81 if the enzyme is exhibiting positive cooperativity, a term that means that the substrate and enzyme bind in such a way that the rate increases to a greater extent with increasing [S] than the MichaeHs—Menten model predicts. Cases with R-s > 81 indicate negative cooperativity so that the catalytic effect becomes less than that found in MichaeHs—Menten kinetics. In these cases, kinetic analysis is usually carried out by means of the HiU equation. 

A large family of enzymes that deviate from hyperbolic kinetics (Michaelis) is the allosteric enzymes. These enzymes contain two or more topologically distinct binding sites that interact functionally with each other. Most commonly, sigmoidal or S-shaped curves are obtained, being indicative of positive substrate cooperativity. The reaction rate for these enzymes can be calculated by the Hill equation ... 

Rates of radial growth of spherulites, constant with time, were highest for pure PCL and decreased with increasing SAN content and with increasing temperature in the range 34-50 C [1061 he. the presence of SAN decreased the rate of crystallisation. The variation in extent of crystallinity with time was sigmoidal and the kinetics of crystallisation were consistent with the Avrami equation (Eq. 26) with an exponent of 3 0.02, consistent with three-dimensional growth... 

The and values can also be determined from a sigmoidal voltammogram obtained under complete kinetic control. Using eqnation (15.23b), one can obtain X from the of such a curve and use it to calculate the sum of two rate constants, k. Then, equation (15.23a) can be used to find K from the plateau current. With both X and A known, the calculation of and is straightforward. 

The relative degree of the crystallinity (A"t) at various crystallization times can be calculated from the ratio of the area of the exothermic peak up to time t divided by that of the total exotherms of the crystallization, and all isotherms exhibited a sigmoidal dependence on time. The time for completing the crystallization was reduced with increasing CNT content, and the relative degree of the crystallinity for the PEN/CNT nanocomposites was higher than that of pure PEN. The crystallization rate of the PEN/CNT nanocomposites was decreased with the isothermal crystallization temperature. The isothermal crystallization kinetics of the PEN/CNT nanocomposites was analyzed using the Avrami equation [68-70] ... 

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