Simple axis of symmetry

One simple practical method of assessing the possibility of the existence of non-superimposable mirror images, particularly with complex structures, is to construct models of the two molecules. The property of chirality may alternatively be described in terms of the symmetry elements of the molecule. If there is a lack of all elements of symmetry (i.e. a simple axis, a centre, a plane, or an n-fold alternating axis) the chiral molecule is asymmetric, and will possess two non-superimposable mirror image structures (e.g. 2a and 2b). If, however, the molecule possesses a simple axis of symmetry (usually a C2 axis) but no other symmetry elements, the chiral molecule is dissymmetric. Thus 4a and 4b are dissymmetric and the simple C2 axis of symmetry, of for example 4a, is shown below. If the molecule possesses a centre of symmetry (C.) or a plane of symmetry (alternating axis of symmetry (S ), the mirror images of the molecule are superimposable and the molecule is optically inactive. These latter three symmetry elements are illustrated in the case of the molecule 4c. 

Enantiotopic ligands and faces are not interchangeable by operation of a symmetry element of the first kind (Cn, simple axis of symmetry) but must be interchangeable by operation of a symmetry element of the second kind (cr, plane of symmetry i, center of symmetry or S , alternating axis of symmetry). (It follows that, since chiral molecules cannot contain a symmetry element of the second kind, there can be no enantiotopic ligands or faces in chiral molecules. Nor, for different reasons, can such ligands or faces occur in linear molecules, QJV or Aoh )... 

There are three symmetry operations each involving a symmetry element rotation about a simple axis of symmetry (C ), reflection through a plane of symmetry (a), and inversion through a center of symmetry (i). More rigorously, symmetry operations may be described under two headings Cn and Sn. The latter is rotation... 

Note that in a number of the molecular structures shown in Figs. 9.1 and 9.2, there is a common element of symmetry. Since all of them are chiral, they do not possess any alternating axis of symmetry, but may have a simple axis of symmetry. As can be easily seen, even from the two-dimensional drawings, 9.8, the ligand of 9.9, 9.10, 9.11, 9.13, and 9.18 have C2 axes of symmetry. 

An example of a twofold simple axis of symmetry (C2 axis) can be seen in planar (S,S)-1, 2-dimethylcyclopentane (1) (Figure 2), which can be brought to an indistinguishable position by a 180° rotation around this axis. As the R-S notation attached to the two asymmetric carbon atoms indicates, these centers have the same chirality in general a C axis may have n identical asymmetric (/IS) units disposed 2n/n apart around this axis. 

Figure 2. C2 Simple axis of symmetry and S4 alternating axis of symmetry.

Thus, dissymmetric molecules commonly have a simple axis of symmetry, and in asymmetric molecules this axis is absent however, both species are usually optically active. In liquid crystal systems both types of material are capable of exhibiting chiral properties. Table 1 summarizes the relationships between optical activity, molecular structure, and rotational symmetry operations [1]. 

Disks fall into the second category of electrode, at which nonlinear diffusion occurs. The lines of flux to a disk electrode (Figure 12.2B) do not coincide with the simple geometries for which we derived Fick s second law, and the diffusion problem must therefore be expressed in two dimensions. Note that a line passing through the center of the disk and normal to the plane of the disk is a cylindrical axis of symmetry, so it is sensible to choose the radial distance from this axis as one of the coordinates for the problem. Diffusion along this radial coordinate, r, is described by Equation 12.7. Also, diffusion along the coordinate, x, normal to the plane of the electrode is described by Equation 12.4. Thus, the form of Fick s second law that must be solved for the disk is ... 

FLC phases in the surface stabilized geometry possess a single C2 axis of symmetry, and therefore polar order and non-zero x<2) in the simple electronic dipolar model. Thus, it is not surprising that experiments aimed at measuring this property were first reported shortly after the Clark-Lagerwall invention. Early studies (14-15) described second harmonic generation in (S)-2-Methylbutyl 4-(4-decyloxybenzylideneamino)cin-namate, the first ferroelectric liquid crystal, also known as DOBAMBC (1). 

This structure has two chiral centres, so how will we know which diastereoisomer we have The answer was simple the stereochemistry has to be tram because Feist s acid is chiral it can be resolved (see later in this chapter) into two enantiomers, Now, the cis diacid would have a plane of symmetry, and so would be achiral—it would be a meso compound, The trans acid on the other hand is chiral— it has only an axis of symmetry. If you do not see this, try superimposing it on its mirror image. You will find that you cannot. 

There are two types of molecular symmetry that cause chemical-shift equivalence. Nuclei or groups of nuclei that are interchangeable by a symmetry operation involving a simple n-fold axis of symmetry (Cn) have been termed equivalent, and are isochronous in chiral and... 

A simple, two-fold axis of symmetry (C2), and, hence, equivalence of the protons interchanged by this symmetry operation, is quite common in specific conformations of inositols and their derivatives, for example, the half-chair conformation (60) of myo-inosose-2 phe-nylosotriazole derivatives,187 and in derivatives of alditols having an even number of chain carbon atoms,188 for example, 2,3 4,5-dianhydro-D-iditol and its 1,6-diacetate and -benzoate189 (61). The 2,5-O-meth-... 

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric. For example, we could obtain a solution for the sedimentation of any axisymmetric body in the direction parallel to its axis of symmetry, but we could not solve for the translational motion in any other direction (e.g., an ellipsoid of revolution that is oriented so that its axis of rotational symmetry is oriented perpendicular to the direction of motion). Another example is the motion of a sphere in a simple linear shear flow. Although the undisturbed flow is 2D and the body is axisymmetric, the resulting flow field is fully 3D. Clearly, it is extremely important to develop a more general solution procedure that can be applied to fully 3D creeping-flow problems. 

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