Amount transformation

Figure 1. Plots showing the Calibration Process. A. Response transformation to constant variance Examples showing a. too little, b. appropriate, and c. too much transformation power. B. Amount Transformation in conforming to a (linear) model. C. Construction of p. confidence bands about the regressed line, q. response error bounds and intersection of these to determine r. the estimated amount interval.

Amount Transformation. Step 2. The amount transformation was performed in a way similar to that of response by use of a power series but for a different reason. In this case linearity was desired in order to use a simple linear regression model. This transformation therefore required a test for satisfactory conformity. One can use a variety of criteria including the correlation coefficient or visual examination of the plot of rgsiduals verses amount. We chose the F test for lack of fit,... 

The amount transformation process is illustrated with data for chlorpyrifos in the flame photometric detector, phosphorus mode, and shown in Table VI. Level 1 transformations were calculated where the amount power was increased by 0.03 units for each step. At an amount power of 0.20 the F statistic of 32.7 showed a minimum but at a confidence level of 95% did not satisfy the F test for linearity. Power steps changed by only 0.01 and 0.001 units in the vicinity of the minimum were then calculated as shown in levels 2 and 3. The best linearity was found in this case at a power transformation of 0.182 although the F statistic of 8.33 did not indicate linearity when compared with the critical F of 2.99 at P=.95. Calculations at these second and third levels were not always necessary and even when performed did not always lead to a satisfactory condition of linearity. 

Table VI. Convergence of the Optimal Amount Transformation for the Determination of Data Linearity. Chlorpyrifos Data with Flame Photometric Detection.

Table VII. The Power for the Amount Transformation Achieving the Best Linearity on Chromatographic Data.

Isothermal Transformation Diagram. To separate the effects of transformation temperature from those of heat flow, it is essential to understand the nature of the transformation of austenite at a given, preselected temperature below the A. Information needed includes the starting time, the amount transformed as a function of time, and the time for complete transformation. A convenient way to accomplish this is to form austenite in specimens so thin (usually about 1-mm thick) that heat flow is not an issue, rapidly transfer the specimens to a Hquid bath at the desired temperature, and foUow the transformation with time. The experiment is repeated at several other transformation temperatures. On the same specimens, the microstmcture and properties of the transformation products can be assessed. These data can be summarized on a single graph of transformation temperature versus time known as an isothermal transformation (IT) diagram or, more usually, a time—temperature—transformation (ITT) diagram. A log scale is used for... 

In Figure 17.10, transformation 39 and 40 constitute examples of nonrotative astereotopomutation, (Class 3A) the former example is stereoaselective, while the latter one is nonstereoselective. As indicated in this figure, there is no rotative achirostereotopomutation (Class 3B). In Figure 17.11, transformations 41 and 42 exemplify nonrotative nonstereotopomutation (Class 4A) the former transformation is stereoaselective (234=235=236=237) the latter one can be either nonstereoselective (if 240(=241) and 242(=243) are formed in equal amounts) or stereoselective (if 240 and 242 are formed in imequal amounts). Transformations 43 and 44 are examples of rotative nonstereotopomutation (Class 4B). Transformation 43 may be either nonstereoselective (if 246(=247) and 248(=249) are formed in equal amounts) or stereoselective (if 246 and 248 are formed in unequal amounts). Transformation 44 is stereoaselective (252=253=254=255). 

All transformations in Figure 17.15 exemplify nonrotative chirostereotopogenesis (Class 7A). Transformations 54 and 55 are cases of sp nonrotative chirostereotopogenesis (As=2 for each component in quartets 312-315 and 318-321.). The former transformation is nonstereoselective 312(=314) and 313(=315) are enantiomers, and are formed in equal amounts. Transformation 55 can be stereoselective, if diastereomers 318(=319) and 320(=321) are formed in unequal amounts if the products are formed in accidentally-equal amounts, the transformation would be considered nonstereoselective. 

Transformations 61 and 62 represent sp rotative chirostereotopogeneses (As=l for each component in quartets 354-357 and 360-363). Each one of these two transformations can be nonstereoselective (if diastereomers 354(=357) and 355(=356) are formed in equal amounts) or stereoselective (if the latter diastereomers are formed in unequal amounts). Transformation 63 is a composite case of sp and sp rotative chirostereotopogeneses (As=2 for each component in quartet 366-369). This transformation is nonstereoselective (if diastereomers 366(=369) and 367(=368) are formed in equal amounts) or stereoselective (if the diastereomers in question are formed in unequal amounts). 

Finally, Figure 18.19 shows three cases of sp rotative stereochirotopogenesis (transformations 103-105), and six cases of sp rotative stereochirotopogenesis (transformations 106-111). Of the sp variety, 103 and 104 are stereoselective (or accidentally nonstereoselective). In transformation 103, stereoselectivity prevails if [525(=528)] [526(=527)] it does not, if the diastereomers in question are formed in equal amounts. Transformation 104 is stereoselective, if diastereomers 531(=534) and 532(=533) are formed in imequal amounts, or nonstereoselective, if the diastereomers are formed in equal amounts. Transformation 105 can be doubly stereoselective (the product mixture consists of imequal amoimts of the four diastereomeric products 537,538,539,540) or stereoselective-and-nonstereoselective (if two of the four diastereomers are formed in accidentally equal amoimts), or, doubly nonstereoselective (if, fortuitously, all four diastereomers are formed in equal amounts). 

Of the sp cases, 106 is nonstereoselective (543(=546) and 544(=545) constitute a racemate). Transformations 107, 109, 110 can be either stereoselective, or accidentally nonstereoselective. Transformation 107 would be stereoselective, if diastereomers 549(=550) and 551(=552) are formed in unequal amounts, or, nonstereoselective, if the two diastereomers in question are formed in equal amounts. Transformation 109 gives two astereomeric diastereomers (diastereomeric mixture 561+563 is astereomeric with respect to diastereomeric mixture 562+564) the transformation can be two-fold stereoselective if, for each of the astereomeric pairs, the diastereomers are formed in unequal amounts ([561] [563] and [562] [564]). If one pair of diastereomers is accidentally formed in equal amounts ([56l]5t[563] and [562]=[564], or vice versa), the transformation may still be stereoselective with respect to the second diastereomeric pair. In principle, the transformation may be two-fold nonstereoselective, if both pairs of diastereomers are formed in pairwise equal amounts ([561]=[563] and [562]=[564]). In the case... 

It may be noted that the implicit assumption that the initial concentration of ammonia would not be significantly reduced by either (a) the amount consumed in complexation with Ag or (b) the amount transformed to NH4, is justified, since neither (a) nor (b) involves as much as 5% of the initial concentration. 

The dependence of the nucleation on time and temperature can be presented in a lucid form by the time-temperature-transformation (TTT) diagram. The amount transformed corresponds to the crystal density All time-temperature conditions leading to the same crystal density are connected by lines. Figure 2.11 shows the TTT diagram derived from the data of Fig. 2.9. The density line p = 1 is the outermost boundary of the nucleation area. It indicates the conditions for the creation of the first effective nucleus. The characteristic shape of the density curves results from the interplay of the two different mechanisms mentioned above, which are responsible for the limitations of the nucleation area at high and low temperatures, respectively. This shape delineates a temperature range of minimum times for the formation of the first nucleus. 

Fig. 10.9 Plot of the relative amount transformed of long chain branched polyethylene against log time for indicated crystallization temperatures. Crystallization temperatures 108.11 °C T 106.72 °C V 105.72 °C 104.1 °C o 101.88 °C. Data from Fig. 10.2. Derived Avrami equation — n = 3 ------------n = 4.

Fig. 10.10 Plot of relative amount transformed for an ethylene-octene random type copolymer, (0.21 mol percent octene) as a function of log time crystallized at 115 °C. (Data from Akpalu et al. (11))...

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