Transformations known with certainty

This algebraic system is diagrammed in Figure 1.2. The input to the system is the independent variable x. The output from the system is the dependent variable y. The transform that relates the output to the input is the well defined mathematical relationship given in Equation 1.1. The mathematical equation transforms a given value of the input, x, into an output value, y. If x = 0, then y = 2. If x = 5, then y = 7, and so on. In this simple system, the transform is known with certainty. 

Figure 1.3 is a system view of a wine-making process. In this system, there are two inputs (yeast and fruit), two outputs (percent alcohol and bouquet ), and a transform that is probably not known with certainty. 

Now let us consider the wine-making example and ask, Can we control temperature to produce a product of any exactly specified alcohol content (up to, say, 12%) The answer is that we probably cannot, and the reason is that the system behavior is initially not known with certainty. We know that some transform relating the alcohol content to the important factors must exist - that is. 

However, the transform W is probably unknown or not known with certainty, and therefore the inverse of the transform (W" ) cannot be known, or cannot be known... 

In 1821 Wohler discovered that a solid deposited from concentrated aqueous solutions of thiocyanic acid. The solid, which was called isoperthiocyanic acid (3-imino-5-mercapto-1,2,4-dithiazole) (361), formed a new product perthiocyanic acid (3,5-dimercapto-l,2,4-thiadiazole) (18) when treated with alkali and then acid. On storage perthiocyanic acid (18) readily reverted to isoperthiocyanic acid (361) (65AHC(5)119). The mechanisms of these interconversions are still not known with certainty but the transformations outlined in Scheme 130 are suggested. Wohler proposed the initial formation of a dimer of thiocyanic acid for which structure (359) appears resonable. Addition of the imine function of (359) to the nitrile function of HSCN would produce the trimer (360) which could readily eliminate hydrogen cyanide to produce isoperthiocyanic acid (361). 

Owing to sluggish phase transformations, the temperature regions of stability for the different modifications are not known with certainty. The S-rh. modification is, however, stable from the melting point of boron, 2573 K, down to 1473 to 1673 K. The -tetr. modification (also denoted tetr. II boron) is stable in the intermediate range down to approximately 1300 K. a-rh. boron is a low-temperature modification, stable up to approximately 1300K. 

The constitution of resazurin is not known with certainty. It contains one oxygen atom more than resorufin, and the two are evidently closely related from the ease with which resazurin may be transformed into resorufin. The two also yield the same dioxyphenazoxin on reduction. It is probable that resazurin is expressed by the formula —... 

The concentration of the functional groups in a polymeric reagent used for synthesis has to be determined so that the maximum amount of reagent available in the polymer can be known with certainty. For a polymer-bound substrate (or its transformed product), the determination must be carried out to discover either the loading of the resin or the yield in a reaction. 

In order to conduct these analyses, the detection limit of the instrument must be known. The detection limit is defined (in ppm or ppb) as the concentration of analyte that allows a detectable signal to be measured with certainty - for example, three times the standard deviation of the background signal or the blank. If the volume of solution needed to obtain these results is known, the preceding values can be transformed to the absolute quantities or mole fractions (pico-mole, femtomole, etc.) that are needed to obtain the signal. In general, these values are excessively small because current instruments use excessively small volumes. 

The use of a probe molecule to study the properties of micellar medium requires the knowledge of the micellar binding site of the probe molecule, with absolute certainty under a specific condition, which is almost impossible to achieve by any experimental or theoretical technique. Pyrene, a highly hydrophobic water insoluble hydrocarbon molecule, has been used to determine the polarity of micellar medium, but its precise micellar binding site is not known with absolute certainty. For instance, pulsed Fourier transform NMR and chemical shift analyses used to determine the dynamic solubilization site of the pyrene in cationic micellar solutions show that pyrene is solubilized in the interior of the... 

The possibility of suspended transformation or the non-formation of the new phases must also be granted in the case where the vapour pressure is raised above that corresponding to the system hydrate— anhydrous salt (or lower hydrate)— vapour in this case the formation of the higher hydrate becomes a possibility, but not a certainty. Although there is no example of this known in the case of hydrated salts, the suspension of the transformation has been observed in the case of the compounds of ammonia with the metal chlorides (p. 76). Horstmann, for example, found that the pressure of ammonia in... 

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