Fraction equivalent

The sample is distilled at predetermined and precisely controlled temperatures under conditions that give a fractionation equivalent to about one theoretical plate. 

Figure 2. Bleaching of merocyanine dye absorption on addition of fractional equivalents of trifluoroacetic acid.

The website http //matti.usu.edu/nlvm/nav/index.html has some helpful interactive exercises that explore the concept of fractional ordering and addition. Upon entering the website, click on Virtual Library. Click on the 6-8 box in the row entitled Numbers Operations. There are several activities to choose from that deal with fractions, including Fraction Pieces, Fractions—Adding, Fractions—Comparing, and Fractions—Equivalent. 

The decimal system is a way to name numbers based on the powers of 10. The numbers to the right of the decimal point are fractional equivalents with denominators that are powers of ten. 

To change a terminating decimal to its fractional equivalent, create a fraction that has all the digits to the right of the decimal point in the numerator and, in the denominator, a power of 10 that has as many zeros as there are digits in the numerator. Then reduce the fraction. For the decimal 0.1875, you write 1875 in the numerator and a 1 followed by four zeroes in the denominator. 

For example, 45 percent has a decimal value of 0.45, and the fraction equivalent is Another example is 0.032 percent, which has a decimal... 

Somewhat greater success has been reported in respect of homo-oligo-merization of olefins, though once again reports are few, and focused predominantly on nickel. The complexes TpxNiCl (Tpx = TpMs 13, TpMs 14, TpMs 15 Ms = mesityl, Tpw = HB(pzN4s)2(pz5Ms), TpMs = HB(pzMs) (pz5 ) were screened in the presence of methylaluminoxane (MAO) cocatalyst at 0 °C under 30 bar ethylene. Under these conditions, both 13 and 14 showed appreciable activity, with very high selectivity for 1-butene (95-96%) within the C4 fraction, equivalent to 81% selectivity overall.13 In contrast, 15 was found to be completely inactive under these conditions. 

Tier I assessments can be refined by adjusting for the actual percent of crop treated. The USEPA considers the percent of crop treated of tolerance residue levels a Tier II exposure assessment. This adjustment is made by mathematically decreasing the tolerance value by a fraction equivalent to the percent of crop treated in the United States. Although the percent of crop treated adjustment can reduce exposure dramatically if the triazine is used only on a small percent of the total crops planted, the use of tolerance values still provides a very conservative exposure estimate. In a Tier II assessment, livestock and poultry feed tolerances can be used to estimate the maximum amount of triazine-treated feed ingested by livestock and poultry and to predict the transfer of residues to edible animal products. 

Using Fractions to Represent Real Situations Proper and Improper Fractions Like and Unlike Fractions Comparing Fractions Equivalent Fractions Simplifying Fractions... 

For the reactor inlet temperature to be markedly below 100°C (which avoids the corrosion and plugging problems ), fuel mass fractions equivalent to 25% CH3OH seem to be necessary for the thermal regime of a flame. This value is supported by experiments as well as it is by theory. In a future configuration it can be lowered if the following improvements in comparison to the presented bench-scale reactor are achieved ... 

Table A.l Certain Decimal Fractions and Their Common Fraction Equivalents ...

Decimal Part of Number Common Fraction Equivalent Multiply by Example ... 

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7], This diffusion equation allows one to include explicitly in Frohlich s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion (0rotation about fixed axis in a potential Vo(< >)- We suppose that a uniform field Fi (having been applied to the assembly of dipoles at a time t = oo so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., pFj linear response condition). 

Table 3-1 Some Common Fraction Equivalents to Decimal Fractions... 

From the total number of units (equipment and replacement batteries) that are eliminated, a first fraction equivalent to forty percent of the total quantity is transferred. This represents the preferred mode of elimination. This equipment is then re-introduced at home as a second hand acquisition. The second fraction or sixty percent of the quantity eliminated is available for collection. 

Kummer and Emmett utilized this technique with a doubly promoted iron catalyst. Thus, radioactive and nonradioactive samples of CO were added in succession as two separate fractions to the reduced catalyst. The chemisorbed CO layer was removed by pumping and analyzed for CO. The results showed that the second fraction of added CO tended to desorb first. However, the results also showed that a partial rapid exchange occurred between the two fractions equivalent to about 50 percent of the iron surface even when adsorption was carried out at -196 °C. 

Geeraert and DeSchepper pointed out that triglyceride separations obtained by capillary GLC on polar colunms are based predominantly on molecular weight differences (carbon number determinations), while reversed-phase HPLC fractionations (equivalent carbon number separations) are based on the degree of unsaturation of the triglycerides. They also noted that the optimization of HPLC conditions could refine the separations obtained, and proposed an optimized reversed-phase... 

Experimentally,/ //f /2 so that AJ Af. Thus the critical elongation of the volume element pervaded by a PS sequence appears to play the major role in governing the crystal growth. Indeed, one may write as a criterion for fractionation (equivalent to the criterion for crystal stability)... 

This is the mass fraction equivalent expression for a CSTR, for a single component i. A similar procedure may be carried out on all species components participating in the system, and combined compactly into a single vector expression, resulting in... 

This is the mass fraction equivalent version of the DSR expression given by Equation 4.8. Similar to the discussion in Chapter 4, a coordinate transformation may be performed on the above expression relating the DSR residence time a in terms of V and G. This gives... 

Note that this is analogous to the DSR expression in concentration space. The variable p used in Equation 9.18 is the mass fraction equivalent of a. Note the geometric interpretation of a DSR operating in mass fraction space is also equivalent—a DSR solution trajectory in mass fraction space is resultant from the linear combination of a reactive component Wr(C) and a mixing vector (z° — z). 

The use of mass fractions in AR theory allows designers to investigate a larger set of systems. Developing an understanding for how to transform molar quantities into mass fraction equivalent space is valuable in tackling systems of industrial and economic relevance using AR theory. 

B. Sample preparation for bioassay. The bioassay utilized 250 mL of urine or extracts or fractions equivalent to 250 mL of urine resuspended in 2 mL of acetone or methanol and added to 250 mL of buffer, saline, or urine. 

A fraction is an expression of division, showing that the numerator is divided by the denominator. A proper fraction is one in which the numerator is smaller than the denominator. In an improper fraction, the numerator is the larger number. A decimal or a decimal fraction is a proper fraction in which the denominator is some power of 10. The decimal fraction is determined by carrying out the division of the proper fraction. Examples of proper fractions and their decimal fraction equivalents are shown in the accompanying table. 

An overall catalyst effectiveness factor no can be derived to take into account the effective surface wetting and the external reactant supply. This factor is defined as the ratio of the actual conversion rate and that obtained without diffusion resistance. This overall effectiveness factor may be approximated by the weighted average value for the differently wetted fractions of the pellet surface (Capra et al. [8]). If the wetting situation can be simplified into one wetted and one non-wetted surface fraction, with pellet volume fractions equivalent to surface fractions, then ... 

Scheme 1. Aliquots of these fractions, equivalent to 1 g of liver, were converted into derivatives for GC-MS as described below.

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