Fractional order

Many ceramic crystals can be described in terms of close-packed planes of oxygen atoms (or ions). For example, corundum, AI2O3, can be regarded as consisting of layers of oxide ions, O2-, with small Al3+ ions contained in an ordered fraction of... 

To compare and order fractions, it is best to find a common denominator and compare the numerators as you would compare integers. 

Another convenient way to order fractions is to convert the fractions into decimal equivalents and compare. Decimals are reviewed in the forthcoming sections. 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

Diagnostic of second-order effects is a residual correlation between isotopic ratios corrected for first-order fractionation (Albarede et al. 2004). 

A correlation between isotopic ratios corrected for mass fractionation may reveal (i) rormded or slopping peak tops (ii) second-order fractionation effects. The necessity of a second-order correction should be established by showing that the bias left after a first-order correction still depends smoothly on the mass. 

The frequent occurrence of ordered fractional-coverage adsorption indicates that adsorbate-adsorbate interactions at close range (S 5 A) are often repulsive. Island formation can occur simultaneously, showing that at larger separations these interactions can become attractive. 

Therefore in a first-order process, the rate of drug movement is directly proportional to the amount of drug (A) in the body, which is usually a function of the dose. K is the first-order fractional rate constant with units of liters/time (time-1) and represents the fraction of drug that is transported per unit of time. Thus in a first-order process, the rate of drug movement is proportional to dose but the fraction moved per unit of time is constant and independent of dose. 

We order fractions, decimals, and percents just as we ordered decimals in Chapter 7, because we convert fractions and percents to decimals before comparing and ordering them. Given a set of fractions, decimals, or percents,... 

Summary of Average Ordered Fraction Values Determined... 

Deuterium, moisture, iodine, and bromine sorption have been utilized for investigating the supramolecular structure of cotton and mercerized cotton. The methods have been described elsewhere [251]. Average ordered fractions are given in Table 5.2. 

In some instances, nonaccessibility, or the so-called average ordered fraction, is measured rather than crystallinity. Values vary depending on the size of the probe molecule and its ability to penetrate and be adsorbed in all the disordered regions. It will be noted that the average ordered fraction is relatively close for the deuteration and moisture regain methods. In addition, the average ordered fraction is decreased about 25% by mercerization. 

Average Ordered Fraction (AVF) and Crystallinity (XAL) of Cotton and Mercerized Cotton Determined by Sorption... 

The values for the average ordered fractions in cotton and mercerized cottons determined by the different chemical methods are summarized in Table 5.6. 

GpC and d - pGpC Platinum Complexes In each case a mixture of isomers is obtained which can be separated into three fractions by HPLC (numbered 1, 2, 3 in the elution order). Fractions 1 and 3 contain only one complex each, respectively I and 3, These complexes equilibrate after HPLC separation. This equilibration is slow enough to allow independent spectroscopic investigations of isomers and 3 after collection at low temperature. Fraction 2 contains two isomers Zb For GpC, the GpC [ Pt] - 2a... 

In this definition, C is the ordered fraction, a is the geometrical factor, and Dh is the Debye-Waller factor. All three factors range in value from 0 to 1. 

To explain these quantities, consider the general case of a discrete atom distribution with N different unit cell position vectors /7 = Xja + yjb + zf for the same fluorescent atom, plus an added random distribution of the same atoms. The ordered fraction C is the fraction of the atoms in the distribution that are coherently located or are crystallographically registered with the substrate crystal lattice. If the atoms occupation fractions for the ordered positions are Cj, C2, Cv, respectively, the ordered fraction is... 

Table I. Average Ordered Fraction Present in Celluloses Determined by Various Techniques ...

Formylation. The fornylation method is based on the determination of the ratio of the extent of esterification of cellulose by formic acid after a given length of time, to that of soluble starch for the same length of time. It is assumed that the starch is completely accessible to the reagent thus a measure of the accessible fraction of the cellulose can be calculated ( ,10). By extrapolating the plot of this ratio against time to zero Hme, the initial accessible fraction of the sample can be determined (2). The ccti )lement of this value is the ordered fraction. Other workers have arbitrarily measured accessibilities after 16 hrs... 

The fractional integral-differential operators (fractional calculus) present a generalization of integration and derivation to noninteger order (fractional) operators. Fust, one can generalize the differential and integral operators into one fundamental Df operator t, which is known as fractional calculus ... 

Da is resolved by rotating the specimen continuously in the X-ray scan over the range 10-65° and traces resolved into peaks (ordered fraction) and background (disordered... 

Although we will usually work with simple integer values for reactant orders, fractional values can occur. For fractional orders the value cannot be determined quite so simply. In the previous example, if doubling the concentration of [A] changed the rate by a factor of 2.83 we would have ... 

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