Mixed-order kinetics

The first kinetic investion of the ethane-1,2-diol reaction was carried out by Price and Knell . However, these workers did not detect the typical mixed-order kinetics which were later established by Duke °, and investigated over a wide range of conditions by Buist and Bunton With excess diol Duke showed that the reaction is first-order with respect to periodate, and that the first-order rate coefficient k is a function of diol concentration, viz. 

For ethane-1,2-diol the diester C2 is in equilibrium with the reactants, and its decomposition to the reaction products is rate limiting and not subject to acid-base catalysis. When the concentrations employed are such that C2 is present in appreciable concentration, the mixed-order kinetics described in section 1.3.2 are observed. Second-order kinetics (for the overall reaction) can arise in three ways (a) C2 in equilibrium, but its concentration negligible, (b) formation of C2 from Ci rate limiting, and the latter in equilibrium with the reactants but present in very low concentration, and (c) formation of Q rate-limiting. For pinacol in the range pH 2-10 alternative (a) cannot be operative because general acid-base catalysis is observed. The most likely step to be subject to catalysis is the formation of C2 from Cl, i.e. alternative (b), because this is a cyclisation and a base (B) could well facilitate reaction by removal of the C-OH proton, viz. 

For the [Co(sep)] +...r system, intermediate h anion decay with mixed-order kinetics was found and accounted for by the concurrent reactions ... 

Keeping the factors such as pH, temperature and enzyme concentration at optimum levels, if the substrate concentration is increased, the velocity of the reaction recorded a rectangular hyperbola. At very low substrate concentration the initial reaction velocity (v) is nearly proportional to the substrate concentration (first order kinetics). However, if the substrate concentration is increased the rate of increase slows down (mixed order kinetics). With a further increase in the subshate concentration the reaction rate approaches a constant (zero order-reaction where velocity is independent of substrate concentration). 

Mixed order kinetics indicate that two or more reactions are taking place in parallel. 

The direction of radical reactions that involve competitive, mixed-order kinetics could be changed as dose rate is increased. Bimolecular reactions would be favored in competition with unimolecular (or pseudo-unimolecular) reactions, because of the higher instantaneous radical concentrations. This effect could be encountered in lipids and in polymers. It is responsible for the greater retention of vitamin B] (thiamin) in pork when irradiated with electron beams than when irradiated with gamma rays (Figure 1) [6]. 

The effect of the addition of small quantities of sulphuric acid upon the reaction order with respect to substrate concentration was investigated. For toluene nitration in aqueous acetic acid, the addition of up to 0.218 mole l i of sulphuric acid increased the first-order rate constant greatly, but did not cause a reversion to zeroth order or mixed order kinetics. The same concentration of sulphuric acid disturbed the first-order nitration in acetonitrile of the more reactive... 

Experimentally it has usually been found that anion-catalyzed reactions are first order with respect to iron(II) (8, 13, 15, 33, 34, 35). The reaction in very concentrated hydrochloric acid solutions (27) should be included in this class. Uncatalyzed reactions have usually been reported to be second order in iron(II) 14, 19, 24), although King and Davidson (15) observed mixed-order kinetics at high temperatures. In no case has the influence of iron (III) been carefully investigated, although Pound (29) reported that addition of iron (III) had no influence on the oxidation rate of iron (II) sulfate. 

Capacity-limited metabolism is also called saturable metabolism, Michaelis-Menten kinetics or mixed-order kinetics. The process of enzymatic metabolism of drugs may be explained by the relationship depicted below... 

In the electron-transfer system in the solvent mixtures with moderate polarities, the back electron-transfer process by the mixed-order kinetics has been observed, suggesting selective solvation of the radical ions, because of large fullerene molecules. 

For nitration carried out in sulpholan ([HNO3] = 4-91 mol 1" ), zeroth-order nitration was observed with mesitylene. With toluene and benzene the kinetics were of mixed-order and first-order, respectively. ... 

With mixed acid 1-methylpyrazole 2-oxide (268) gives high yields of l-methyl-5-nitropyrazole 2-oxide (269) (B-76MI40402). The form undergoing the reaction is the base, for which first-order kinetics are observed in the Hq range from -5 to -6.5 a dinitro derivative is also formed under slightly different conditions. The reaction of indazole with fuming nitric acid affords a nearly quantitative yield of 5-nitroindazole (Section 4.04.2.3.2(i)). 

Kinetic data" for a reaction that follows mixed second-order kinetics EtO + EtMe2SI = Et20 + Me2S + I ... 

From this, the values of [B], follow from Eq. (2-17). This equation can also be used to fit the data with a nonlinear least-squares routine. Table 2-2 gives an example of data for a reaction that follows mixed second-order kinetics.3 Figure 2-3 displays the linear variation of ln([B],/tA],) with time as well as [A], itself against time. Both show a line corresponding to the least-squares fit of the function given. 

Kinetic data for the reaction between PuOi- and Fe2+, given in Table 2-4, are fitted to the integrated rate law for mixed second-order kinetics. The solid curve represents the least-squares fit to Eq. (2-34). left and (2-35). right. 

Mixed second-order kinetics. Consider an electron transfer reaction between the triva-lent ions of neptunium and iron in aqueous solution 23... 

Saddle point. 170 Salt effects. 206-214 Scavenging (see Reactions, trapping) Second-order kinetics. 18-22, 24 in one component, 18-19 in two components (mixed), 19-22 Selectivity. 112 Sensitivity analysis. 118 Sensitivity factor, 239-240 Sequential reactions (see Consecutive reactions)... 

A second investigation included an observation of an increase in absorption at 470 nm on mixing the reactants suggesting fast complex formation this absorption then decayed with first-order kinetics and with a rate linearly dependent upon hydroquinone concentration. k2 decreases with increasing acidity in a manner indicating the complex between MnOH " and hydroquinone to be the exclusive reactant. E2 = 14.0+0.7 kcal.mole and AS = 7.5+ 2.1 eu. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

Toth et al. then used laser flash photolysis as a means to determine the value of k x independently of the above study (8). They used 355 nm laser light to photolyze mixtures of C102 and Br2/Br s. Absorption of this light by Br3 led to the prompt formation of Br2, and the subsequent loss of Br2 was monitored by its absorbance at 360 nm. The loss of Br2 occurred with mixed 2nd- and lst-order kinetics due to the parallel 2nd-order self reaction of Br2 and its pseudo-first-order reaction with C102. These experiments led to a value of 3.6 x 109 M 1 s 1 for kh which is in good agreement with the approximate value (1.1 x 109 M 1 s ) originally obtained. 

Laboratory and mixed field/laboratory studies have confirmed that half-order kinetics for DO surface removal rates may be a reasonable approximation for sewer biofilm (Raunkjaeretal., 1997 Bjerre etal., 1998b). These results also showed the influence of readily biodegradable substrate. Furthermore, temperature dependency limited by diffusion is included (Nielsen et al., 1998). The following equation for the aerobic growth rate was therefore used ... 

It was found that three factors affect the biotransformation rates of A9PEO the origin of the bacterial culture, temperature, and the initial concentration of A9PE0n. Biotransformation kinetics of mixed bacterial cultures from the brackish water layer was faster than those from the saline water layer. Rate constants (based on first order kinetics) for... 

RP-HPLC found application in the monitoring of the alkali hydrolysis kinetics of alkali-clearable azo disperse dyes containing a fluorosulphonyl group. The chemical structures of dyes included in the experiments are shown in Fig. 3.85. Samples for RP-HPLC analysis were neutralized to pH 4.0 - 4.5 with diluted HC1 mixed with five volumes of ACN and injected without any other sample preparation step. Separation was carried out in an ODS column at ambient temperature. The isocratic mobile phase consisted of ACN-water (80 20, v/v) and dyes were detected at their absorption maxima. HPLC measurements indicated that dyes are easily hydrolysed under relatively mild alkaline conditions, and the hydrolysis follows a pseudo first-order kinetics [148],... 

Curvature may result when kinetic data are plotted. This may be due to an incorrect assumption of reaction order. If first-order kinetics is assumed and the reaction is really second order, downward curvature is observed. If second-order kinetics is assumed but the reaction is first-order, upward curvature is observed. Curvature can also be due to fractional, third, higher, or mixed reaction orders. Non-attainment of equilibrium often results in downward curvature. Temperature changes during the study can also cause curvature thus, it is important for temperature to be controlled accurately during a kinetic experiment. 

In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

This behavior is sometimes referred to as saturation kinetics. When ft [B] < 1, the observed second-order is easily understood (rate constant = a). When 6 [B] 1 there is a mixed-order... 

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