Fractional-order kinetics

Agrawal and Wei (1984) isolated the metallochlorin and confirmed that the apparent fractional order kinetics resulted from a sequential mechanism much like HDS and HDN reactions. The hydrodemetallation of both nickel- and vanadyl-etioporphyrins on oxide CoMo/A1203 proceeded through two mechanistically different steps, an initial reversible hydrogenation followed by a terminal hydrogenolysis step... 

A review of the literature shows that photochemical reactions generally follow first-order kinetics. However, pseudo first-order, zero-order, and even fractional-order kinetics have been reported. Drugs reported to follow first-order kinetics include nifedipine (8) doxepin (41), riboflavin (28), minoxidil (48), adriamycin (52), doxorubicin, daunorubicin and epirubicin (61), folic acid (66), menadione sodium bisulfite (67), tetracycline hydrochloride (71), decarbazine (73), furosemide (74), democlocycline (76) and hydrocortisone, and prednisolone (90). Apparent pseudo first-order rate has been reported for pyridoxine (84). Indomethacin (91), sodium nalidixate (65), methotrexate (92), and metronidazole (68) have been reported... 

Very often, common natural processes involving diffusion and reaction are found to obey power laws which for most of the time have been described within the domain of Euclidean space and hence restricted to integer powers. Table 2. gives a comparison of the Euclidean and fractal geometries. On the other hand, it is observed that a large number of heterogeneous reactions follow fractional-order kinetics under different process conditions [13]. But most classical transport theories, valid for Euclidean structures, fail when applied to transport processes... 

Interpret mixed and fractional order kinetic behaviour... 

To avoid the restrictions imposed when desorption can only occur from the precursor state, Le Lay et al. [279] allowed for a direct desorption flux, which of course always gives fractional-order kinetics within the framework of this model if this process is rate-controlling. Le Lay et al. thus defined two possible rate limitations, the zero-order case when desorption occurs from a steady state precursor population and fractional order with direct desorption. 

Oxidation of AA with methylene green (MG) is accelerated when atmospheric oxygen is replaced with carbon dioxide. The reaction followed fractional order kinetics... 

Sterilization of Media First-order kinetics may be assumed for heat destruction of living matter, and this leads to a linear relationship when logarithm of the fraction surviving is plotted against time. However, nonlogarithmic kinetics of death are quite often found for bacterial spores. One model for such behavior assumes inactivation of spores via a sensitive intermediate state by the mechanism ... 

These two parameters describe the change in fraction unconverted with a percentage change in kt or in c0. The first sensitivity is also the slope of the curves in Fig. 28. The values of these sensitivities are given in Table IX. In a piston flow reactor where the conversion level is c/c0 = 0.1, the value of Stt is —0.23 for the first-order kinetics, —0.90 for the zero-order kinetics, and —4.95 for the negative first-order kinetics. In the stirred tank reactor, the value of the sensitivities Skt is —0.09 for the first-order kinetics, — 0.90 for the zero-order kinetics, and +0.11 for the negative first-order kinetics. A positive sensitivity means that as kt is increased, the fraction unconverted also increases, clearly an unstable situation. 

In deducing from the resulting kinetic equation the nature of the electrophile and how it is produced it is important to represent all the reagents present in terms of the species which they may produce. In this way it is possible to eliminate many negative or fractional orders in reagent and generally obtain a simpler kinetic equation. For example, the observed rate law in the uncatalyzed iodination of aniline can be written21,22 as... 

Fractional orders usually result when no single step in the reaction is solely ratedetermining and intermediate kinetics result. They may also arise if the electrophile is produced by the dissociation of a reagent such that the species produced are not buffered. 

When a reaction follows first-order kinetics, [A] decreases exponentially with time. According to Eq. (2-3), a plot of In [A], against time is a straight line with a slope of -k. This slope and the rate constant are independent of [A]o- This straight line also means that a definite fraction of the reaction occurs over a given time interval, independent of [A]0. If this fraction is taken as 50 percent, substitution of [A] 1/2 = [A]q/2 into Eq. (2-2) gives... 

The decay of the absorption of e, - was followed at 1000 nm30 or 900 nm50 where the oxidizing species do not absorb. It was found30 that esol decays by first-order kinetics. A second-order rate constant was calculated assumi ng that the decay is only by reaction with DMSO. These second-order rate constants appear to go through a maximum between 0.20 and 0.43 mole fraction of DMSO where k = 5.6 x 106 m 1 s , however, there is not a large difference between the different concentrations as the lowest value is 2.9 x 10 ... 

Although many reaction-rate studies do give linear plots, which can therefore be easily interpreted, the results in many other studies are not so simple. In some cases a reaction may be first order at low concentrations but second order at higher concentrations. In other cases, fractional orders as well as negative orders are obtained. The interpretation of complex kinetics often requires much skill and effort. Even where the kinetics are relatively simple, there is often a problem in interpreting the data because of the difficulty of obtaining precise enough measurements. ... 

This gives = 0.8047 h. The molecular weight of the monomer, Ma, is not actually used in the calculation. Extrapolation of the first-order kinetics to a 4-h batch predicts that there will be 900exp(-3.22) = 36kg or 4% by weight of monomer left unreacted. Note that the fraction unreacted, Ya, must be defined as a ratio of moles rather than concentrations because the density varies during the reaction. 

Example 3.5 A 1-in i.d coiled tube, 57 m long, is being used as a tubular reactor. The operating temperature is 973 K. The inlet pressure is 1.068 atm the outlet pressure is 1 atm. The outlet velocity has been measured to be 9.96 m/s. The fluid is mainly steam, but it contains small amounts of an organic compound that decomposes according to first-order kinetics with a half-life of 2.1s at 973 K. Determine the mean residence time and the fractional conversion of the organic. 

Failure to give a product because of diffusion away of a reactant may give rise to kinetic competition between two processes reaction with activation energy E and diffusion with activation energy Ej- This competition can easily be handled using assumed first-order kinetics (for correlated pairs of reactants) and considering the fraction, F, of the available reaction sites which lead to products within infinite time compared to the fraction, — F, which give no reaction—presumably by diffusion away of a reactant. This treatment leads to the expression... 

C15-0058. Radioactive isotopes decay according to first-order kinetics. For one particular isotope, 1.00 nmol registers 1.2x10 decays in 1.00 min. (a) How many decays will occur in 1.00 min if 5.00 nmol of this isotope are present (b) What fraction of the isotope decays per minute in each case (c) Explain the relationship between your answers to (a) and (b). 

Steady-state approximation. Fractional reaction orders may be obtained from kinetic data for complex reactions consisting of elementary steps, although none of these steps are of fractional order. The same applies to reactions taking place on a solid catalyst. The steady-state approximation is very useful for the analysis of the kinetics of such reactions and is illustrated by Example 5.4.2.2a for a solid-catalysed reaction. 

Fractional Order. In the decomposition of pure solids, the kinetics of reactions can often be more complex than simple zero- or first-order processes. Carstensen [88] has reviewed the stability of solids and solid dosage forms as well as the equations that can be used in these cases. In addition to zero- and first-order kinetics, solid-state degradations are often described by fractional-order equations. 

Notice that in the region of fast chemical reaction, the effectiveness factor becomes inversely proportional to the modulus h2. Since h2 is proportional to the square root of the external surface concentration, these two fundamental relations require that for second-order kinetics, the fraction of the catalyst surface that is effective will increase as one moves downstream in an isothermal packed bed reactor. 

The first-order and fractional power kinetics were also used to describe the behavior of DEHP biodegradation in the thermophilic phase, including the initial mesophilic phases (phase I) and the phase thereafter (phase II), respectively [62]. The fractional power kinetic model parameters, i.e., K and N. were calculated by (l)-(3) and derived from a plot of log (C/Co) versus log(f). The half time (f0.s) of DEHP degradation in phases I and II was calculated using first-order and fractional power kinetic equations (3), respectively. 

First-order rate constants are used to describe reactions of the type A — B. In the simple mechanism for enzyme catalysis, the reactions leading away from ES in both directions are of this type. The velocity of ES disappearance by any single pathway (such as the ones labeled k2 and k3) depends on the fraction of ES molecules that have sufficient energy to get across the specific activation barrier (hump) and decompose along a specific route. ES gets this energy from collision with solvent and from thermal motions in ES itself. The velocity of a first-order reaction depends linearly on the amount of ES left at any time. Since velocity has units of molar per minute (M/min) and ES has units of molar (M), the little k (first-order rate constant) must have units of reciprocal minutes (1/min, or min ). Since only one molecule of ES is involved in the reaction, this case is called first-order kinetics. The velocity depends on the substrate concentration raised to the first power (v = /c[A]). 

The rate also varies with butadiene concentration. However, the order of the rate dependence on butadiene concentration is temperature-de-pendent, i.e., a fractional order (0.34) at 30°C and first-order at 50°C (Tables II and III). Cramer s (4, 7) explanation for this temperature effect on the kinetics is that, at 50°C, the insertion reaction to form 4 from 3, although still slow, is no longer rate-determining. Rather, the rate-determining step is the conversion of the hexyl species in 4 into 1,4-hexadiene or the release of hexadiene from the catalyst complex. This interaction involves a hydride transfer from the hexyl ligand to a coordinated butadiene. This transfer should be fast, as indicated by some earlier studies of Rh-catalyzed olefin isomerization reactions (8). The slow release of the hexadiene is therefore attributed to the low concentration of butadiene. Thus, Scheme 2 can be expanded to include complex 6, as shown in Scheme 3. The rate of release of hexadiene depends on the concentra-... 

---