Half-order kinetics

Reaction orders of 5 and are found on occasion. They often (but not always) signal a chain mechanism (see Chapter 8). If the reaction follows half-order kinetics, the equations are... 

Initiation rate. The oxidation of ELPO by S Ojj" follows pseudo-half-order kinetics when [(EPOC] [S Ojj". The plot of k]t versus [ELPOf J is linear [see Eq. (8-46)]. 

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 must be concluded, therefore, that the kinetic scheme proposed in Equations (2) and (3) cannot be valid for these systems. Although the seeming correspondence between the half-order kinetics and the dimeric association in the case of styrene might validate the above kinetic scheme, even this hypothesis has been recently contradicted (18). Thus Fetters and Young... 

Figure 6. Dependence of the polymerization quantum yield ( ) on the light-intensity (Iq) in the laser-induced polymerization of epoxy-acrylate photoresists (— expected variation of on I0 for a half-order kinetic law).

This last explanation appears to be the most feasible since we did not observe deviations from the expected half-order kinetic law when those multiacrylate monomers were polymerized in dilute solution where no rigid network is formed (30). A further feature which corroborates this conclusion is... 

A Fe/Sb = 1/6 catalyst was investigated by Bakshi et al. [36] in a differential flow reactor, and compared with a Sn/Sb catalyst. With both, approximately half-order kinetics are found with respect to both the olefin and oxygen. At product partial pressuresPc4h5 andpH2o below 0.02 atm the rate of butadiene formation is described by... 

Fig. 2. Conversion data from half-order kinetics tested against the kinetics of Topchieva and Panchenkov (7).

It is evident that two distinct types of behaviour can occur depending on the conditions. When the rate of atomisation is slow relative to the collision rate, is proportional to (Fj)-172 (half-order kinetics) and when the rate of atomisation is comparable to the collision rate, approaches a limiting upper value (first-order kinetics). The circumstances in which these two kinds of behaviour are to be expected will now be examined. 

Examination of eqn. (46) shows that the temperature coefficient of Dt, in the region of half-order kinetics, is dominated by the temperature coefficient of Kp. If we write... 

The physical interpretation of the conditions defined by eqn. (44) is that there will be many molecules adsorbed and desorbed for every atom which leaves the surface. This means that the stationary concentration of the adsorbed layer is negligibly different from the equilibrium value. In the region of half-order kinetics, the mechanism of atomisation is... 

Tret yakov [111] did not find half-order kinetics for the atomisation of hydrogen by gold under these conditions. On the contrary, they reported first-order kinetics. They attributed the observation of half-order kinetics by Brennan and Fletcher for this system to the presence of surface impurities capable of rapidly dissociating molecular hydrogen, whereas molecular adsorption on the clean surface, being activated, was considered by them to be rate-determining. We will return to the H2—Au system in Sect. 3.2.1(d). Nomes and Donaldson [8] have demonstrated half-order behaviour for the N2—W system, but over a narrower pressure range. 

Experimental rates of formation (ca) of atoms per unit area of surface in the region of half-order kinetics at 1200 K, expressed in the form ua = b(Pi)X12 exp (— EJRT), Values of si(s2/s2)12 have been calculated from the theoretical equation, eqn. (48), and the values of its parameters given in Table 2. 

Typical conditions for half-order kinetics are T = 1200 K and P 2 = 10 3 torr, for which it is readily verified that the expression in the square brackets is dominated by the term in P 2. Hence... 

They claim, in an entirely qualitative way, that this scheme is compatible with half-order kinetics and an activation energy of D H2), assuming... 

The overall first-order kinetics is thus accounted for. This interpretation of the kinetics, in terms of a three-halves order for reaction (3), was first suggested by Quinn . The initiation and chain-propagating steps were originally suggested in 1934 by Rice and Herzfeld . However, until the matter of the order of reaction (3) became clarified there remained considerably uncertainty about the nature of the chain-ending step. Because reaction (3) is in its intermediate pressure region the ethyl radical is neither a p nor a fi radical, but is about half-way between. First-order initiation with im termination leads to half-order kinetics, and with pp termination to three-halves order if the radicals are half-way between p and /i the result is first-order kinetics. 

The other alkali metals have been less extensively studied. The propagation rates of polystyrylsodium, -potassium, -rubidium and -cesium have been measured in benzene and cyclohexane [72, 73]. The sodium compound still shows half order kinetics in active centre concentration and is presumably associated to dimers. The rates for the rubidium and cesium compounds are directly proportional to the concentrations of the active chains which are presumably unassociated in solution. Absolute kp values can be determined from the propagation rate in this case. Poly-styrylpotassium shows intermediate behaviour (Fig. 11), the reaction order being close to unity at a concentration of the potassium compound near 5 x 10 M and close to one half at concentrations around 10" M. It could be shown by viscosity measurements that association was absent in the low concentration range. In this system both K2 and kp can be measured. The results are summarized in Table 2. The half order reactions show a large increase in kpK between lithium and potassium which... 

This is the equivalent of Eqs. (18) and (20) it correctly describes the observed half-order kinetics and requires an activation of Ei=Ei — E. The other limit of Eq. (28) is reached when... 

The assumption of mechanism (I) requires that the activation energy in the region of half-order kinetics should be — E = 51.5 kcal mole [Eq. (31)]. This is in very good agreement with the result of Brennan and Fletcher (Table II) and the theoretical expectation that the energy of activation Ei should be + i, being less than... 

The comparison of the requirements of Eq. (39) with experiment is a severe test of the validity of the assumption made in the derivation of the equation. In Figs. 2(a) and (b), the experimental data for hydrogen on tungsten and platinum are compared with the theoretical predictions based on filament temperatures calculated from the observed rates in the region of half-order kinetics and the data of Table II. The temperatures obtained in this way are 1830°K for tungsten and 1750°K for platinum (cf. 1800° and 1750°K, respectively, for the nominal temperatures as measured with the optical pyrometer). The agreement between theory and experiment is seen to be very satisfactory over the very wide pres-dure range involved. 

Suppose that the following reaction is a series network with square kinetics for the first reaction and half-order kinetics for the second ... 

What would be the result if the reaction were halforder Repeat the analysis for half-order kinetics, using the same values of the parameters in appropriate units. 

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