Geometrical formulae

The geometrical formula of a compound represents the geometrical arrangement of the atoms in the molecule. It is determined from the structure of the compound. To represent the geometry properly really requires a three-dimensional model, but we have to try to do this in two dimensions. 

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The angle 6 ab between these vectors is determined by the geometrical formula... 

In the original W1/W2 paper [1], we opted for the geometric formula in view of the observed geometric convergence behavior. In a subsequent validation study [26] on a much wider variety of systems, we however found the two-point formula to be much more reliable, and we have adopted it henceforth. 

In special cases, when the geometrical shape of a specimen is simple, its change in volume can be estimated by means of a simple measurement of its dimensions and multiplying these according to geometrical formulas for volume determination. 

Table C.l Geometrical Formulae in Various Surface Representations...

All liquid droplets spontaneously assume the form of a sphere in order to minimize their surface free energy by minimizing their surface area to volume ratio, as we will see in the thermodynamic treatment in Chapter 3. The surface area to volume ratio can easily be calculated for materials having exact geometric shapes by using well-known geometric formulas. For example, for a sphere and a cube this ratio is... 

Tables 3.1-3.3 summarize approximate scaling behavior of tubular reactors operating with fluids having constant physical properties. Many of the factors are merely ratios of geometric formulas. Those for pressure drop are based on Equation 3.18 for laminar flow and Equation 3.19 for turbulent flow. Results for heat transfer are included for convenience. Tables 3.1 and 3.2 are both for laminar flow and are identical except for the heat transfer correlations that are discussed in Chapter 5. Table 3.3 is for turbulent flow. Scaling in parallel is not shown since the scaling factors for a single tube would all be 1.

Total bark surface of 3 McIntosh trees was determined from 1983 studies, by measuring length and circumference of trunk, branches, and twigs <1 cm in and >.5 cm diameter were measured with calipers and a tape measure. Total bark/tree surface area was determined by geometrical formula for a frustum. Both bark, foliage and leaf surface determinations were made within 10 days of application to minimize growth effects. 

Let us consider the determination of molecular characteristics S and for butadiene-styrene rubber. As it is known [15], the value of macromolecule diameter square is equal to for polybutadiene - 20.7 and for polystyrene - 69.8 A. Calculating the macromolecule, simulated as cylinder, cross-sectional area for the indicated polymers according to the known geometrical formulae, let us obtain 16.2 A and 54.8 A, respectively. Further, accepting as S for butadiene-styrene rubber mean value of the cited above areas, let us obtain S=35.5 A. Further the characteristic ratio can be determined, which is a polymer chain statistical flexibility indicator [16], with the aid of the following empirical formula [14] ... 

Until Kekule in 1877, all geometrical formulas referred to the structure and behavior of small molecules. However, Kekule in 1877 during a lecture upon becoming the Rector of the U. of Bonn advanced the hypothesis that the natural organic substances associated with life (proteins, starch, cellulose, etc.) may consist of very long chains and derive their special properties from this peculiar structure (Tolbolsky and Mark, (1971)). 

Numerical Kekule Valence Structures. Kekule valence structures, like Clar s valence structures, may be viewed as geometrical formulas because they are drawn. Similarly, the novel notation for benzenoid hydrocarbons using an inscribed Greek jt represents a pictorial, qualitative, non-numerical representation of such molecules. However, well-designed or selected notation may have advantages that initially one did not anticipate. This has been the case with the Leibniz notation for derivatives in mathematics, and with graphical codes of configuration of /7-alkanes in structural chemistry. We will show now that the just introduced qualitative Jt-nota-... 

For convenience, listings of Selected Physical Constants, Some Common Conversion Factors, Some Useful Geometric Formulas, and Location of Important Data and Other Useful Information are presented on the inside of the back cover. 

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