Order noninteger

Several important points about the rate law are shown in equation A5.4. First, the rate of a reaction may depend on the concentrations of both reactants and products, as well as the concentrations of species that do not appear in the reaction s overall stoichiometry. Species E in equation A5.4, for example, may represent a catalyst. Second, the reaction order for a given species is not necessarily the same as its stoichiometry in the chemical reaction. Reaction orders may be positive, negative, or zero and may take integer or noninteger values. Finally, the overall reaction order is the sum of the individual reaction orders. Thus, the overall reaction order for equation A5.4 isa-l-[3-l-y-l-5-l-8. 

Guideline 11. Whenever a rate law contains noninteger orders, there are intermediates present in the reaction sequence. When one observes a fractional order in an empirical rate expression for a homogeneous reaction, it is often an indication that an important part of the mechanism is the splitting of a molecule into free radicals or ions. 

The reaction orders obtained from nonlinear analysis are usually nonintegers. It is customary to round the values to nearest integers, half-integers, tenths of integers, etc. as may be appropriate. The regression is then repeated with order(s) specified to obtain a revised value of the rate constant, or revised values of the Arrhenius parameters. 

We see that the half-life is always inversely proportional to k and that its dependence on [A]o depends on the reaction order. Thereby the method can be used to determine both the rate constant and the reaction order, even for reactions with noninteger reaction order. Similar to the integral method, the half-life method can be used if concentration data for the reactant are available as a function of time, preferably over several half-lives. Alternatively the half-life can be determined for different initial concentrations in several subsequent experiments. 

Formaldehyde is formed as an intermediate product. This suggested two-step reaction would indicate two second-order reactions occurring in sequence. The over-all apparent order of reaction of Equation 3, if expressed in a form suggested by Equation 2, could be of noninteger order. If one of the reactions suggested by Equations 4 or 5 was extremely fast compared to the other, the reaction would essentially be controlled by the slower one and would appear to be of the second order. Similarly, the standard combustion equation of hydrocarbons with air is usually written as... 

To evaluate the activation energy, E, and the collision frequency factor, A, recourse may be made to Semenov s equation (18). To accommodate noninteger orders of... 

In the case of noninteger spin nuclei, such as 27A1 or 23Na, we observe only the central —transition, as the other transitions are spread over too wide a frequency range. The first- and second-order frequency shifts are... 

The first-order frequency shift vanishes for m = 3, which means that the central transition for noninteger spin nuclei is not affected to first order by quadrupolar interactions (see Fig. 2). It is clearly advantageous to work with these, as the — 1 <- 0 and 0 <-> 1 transitions for integer spin nuclei are always shifted. 

The following strategy therefore emerges for the study of quadrupolar nuclei observe the central transition of nuclei with noninteger spin, use MAS (to remove dipolar coupling, chemical shift anisotropy, and first-order quadrupolar effects), and work at high fields (to minimize second-order effects). 

In the same vein and under dimensionally restricted conditions, the description of the Michaelis-Menten mechanism can be governed by power-law kinetics with kinetic orders with respect to substrate and enzyme given by noninteger powers. Under quasi-steady-state conditions, Savageau [25] defined a fractal Michaelis constant and introduced the fractal rate law. The behavior of this fractal rate law is decidedly different from the traditional Michaelis-Menten rate law ... 

The rate law determined from experiment has a noninteger order ... 

It is acceptable (and usual) for the value of n calculated from Equation (14-12) to be a noninteger in Equation (4-11) to calculate the conversion. For reactions other than first-order, an integer number of reactors must be used and sequential mole balances on each reactor must be carried out. For example, if x /a =2.8 then one would round up to three tanks. The conversion and effluence concentrations would be solved sequentially using the algorithm developed in Chapter 4. That is, after solving for the effluent from the first tank, it would be used and the input to the second tank and so on. 

The majority of the degradation reactions of pharmaceuticals take place at finite rates and are chemical in nature. Solvent, concentration of reactants, temperature, pH of the medium, radiation energy, and the presence of catalysts are important factors that affect these reactions. The order of the reaction is characterized by the manner in which the reaction rate depends on the reactant concentration. The degradation of most pharmaceuticals is classified as zero order, first order, or pseudo-first order, although the compounds may degrade by complicated mechanisms, and the true expression may be of higher order or be complex and noninteger. 

Figure 7.1. Moments of a normal distribution versus moment order. If just integer moments are known (symbols), simple interpolation (continuous line) allows one to estimate noninteger moments.

In traditional chemical kinetics, where reactions are assumed to occur in dilute homogeneous solutions, the kinetic orders are typically integer values that add up to the molecularity of the reaction. The concepts of kinetic order and molecularity are distinct, although they often are confused and considered to be equivalent because of this association in traditional chemical kinetics. However, even within traditional chemical kinetics, it has long been known that reactions involving free radicals can have kinetic orders with noninteger value (Benson, 1960). 

In vivo, where reactions often occur on membranes or in channels, the kinetic order can be expected to exhibit noninteger values that are larger than the number of molecules entering into the reaction, and the forms of the rate laws will be different from those obtained by assuming homogeneous conditions. 

Reaction kinetics, or at least the mathematical formulas describing it, depend on the order of the reaction. Orders of 0, 1, and 2 are mostly considered. Reaction order, however, is an empirical number, mostly not equal to the molecularity of the reaction. It has to be determined experimentally, and noninteger values may be observed moreover, order may change in the course of the reaction or may vary with conditions such as temperature. 

The overtone order may turn into a noninteger number if piezoelectric stiffening and energy trapping are taken into account. This does not change the structure of the equations. 

Senn SJ (2007) Drawbacks to noninteger scoring for ordered categorical data. Biometrics 63 296-299. 

At intermediate pressures, a unimolecular reaction might appear to have a noninteger order, such as 1.3 or 1.75, but such values have no physical significance, and the order is likely to change when the concentration is varied over a wider range. 

FIGURE 13.12 Contour for representation (13.81) of Bessel function J x) of noninteger order. 

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