Shaded-sphere pictures

Shaded-Sphere Pictures. Each atom is shown to represent complete chemical detail of molecules. This category includes three basic types of picture ... 

Figure 1.8. The top left sphere shows the positive (shaded) and negative (unshaded) regions for the real-valued function 2 - The top right sphere shows the pure real (solid) and pure imaginary (dashed) meridian for the function 72,2- The bottom picture shows the zero points (double-dashed) as well as the pure real (soUd) and pure imaginary (dashed) meridians of 12,1 There are colored versions of these pictures available on the internet. See, for instance, [Re].

Fig. 4. Inclusion matrix of Dianin s compound (i) (schematic representation) Individual Dianin molecules are represented by a specified hydroxy group attached to an elUpsoid. The characteristic hydrogen bridge networks are indicated by the shaded hexagons (H-bonds in dotted lines). The hatched sphere in the centre of the cavity pictures an included guest molecule, e.g. chloroform...

From the picture it is clear that the rods may easily avoid each other. For any typical orientation of the first rod, the second rod may have almost any orientation without hitting it. Since all orientations are equally likely, the probability that the second rod would have a forbidden orientation is proportional to the shaded fraction of the sphere in the Figure. This fraction is small thus the probability g r) that two randomly oriented rods would not intersect is near unity. It is much larger than it was for the two clouds. Accordingly, the excluded volume is much smaller than for the clouds. To work out the allowed orientations and thus the excluded volume is a classic exercise [8.15]. The result is that 1 — g r) w/r, compared with 1 for the clouds. The average (1 — g r)) is roughly w/R. Thus 14 w/R)R , compared with R for the clouds. 

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