Enantiomorphous objects

The homochirality problem arises whenever there exists a pathway of continuous deformation that connects two enantiomorphous objects but does not require passage through an achiral point. Under these circumstances, it becomes meaningless to speak of such objects as right-handed or left-handed. The case of the asymmetric tetrahedron is the simplest example. 

Figure 1.9. Trigonal pyramid with its apex up (left) and down (right) relative to the plane of the paper. Shading is used to emphasize enantiomorphous objects.

A chiral object and the opposite object formed by inversion form a pair of enan-tiomorphs. If an enantiomorph is a molecular entity, it is called an enantiomer. An equimolar mixture of enantiomers is a racemate. 

Pasteur thus made the important deduction that the rotation of polarized light caused by different tartaric acid salt crystals was the property of chiral molecules. The (+)- and ( )-tartaric acids were thought to be related as an object to its mirror image in three dimensions. These tartaric acid salts were dissymmetric and enantiomorphous at the molecular level. It was this dissymmetry that provided the power to rotate the polarized light. 

Two equal and similar right hands are homochirally similar. Equal and similar right and left hands are heterochirally similar. .. These are also called enantiomorphs ,. .. Any chiral object and its image in a plane mirror are heterochirally similar". 

Chiral crystals, like any other asymmetric object, exist in two enantiomorphous equienergetic forms, but careful crystallization of the material can induce the entire ensemble of molecules to aggregate into one crystal, of one-handedness, presumably starting from a single nucleus (Figure 2). However, it is not uncommon to find both enantiomorphs present in a given batch of crystals from the same recrystallization. 

Enantiomorph is one of a pair of chiral objects or models that are nonsuperposable mirror images of each other. The adjective enantiomorphic is also applied to mirror image-related groups within a molecular entity. 

Since chirality is a geometrical property, all serious discussions on this topic require a mathematical treatment that is much out of this review. Note, however, that if you cut by the middle of a Klein bottle (an achiral object having a plane of symmetry), you obtain two Mobius strips both chiral and mutually enantiomorph (Fig. 3.5). This pure mathematical result is closely related to the situation of meso compounds described above [11]. 

An object is chiral and a chiroid if and only if it cannot be superposed on its mirror image by a proper congruence, otherwise it is achiral two chiroids are heterochiral and enantiomorphs if and only if they are improperly congruent and two chiroids are homochiral and homomorphs if and only if they are properly congruent. 

The relationship between an object and its mirror image is at the heart of the incongruity of counterparts. In particular, the statement that enantiomorphs can be brought into congruence only by rigid motions combined with an odd number of reflections in the object s space requires further elaboration. 

Objects that exhibit -invariant enantiomorphism are either immobile or undergo a motion that can be reduced to a screw displacement (as exemplified in Figure 4). Objects that exhibit 7-noninvariant enantiomorphism behave effectively like stationary spinning cones (cf. Figure 5). We now show that all these objects belong to chiral groups. 

To visualize an object with Dm symmetry, imagine a cylinder whose outside is covered with n slanted striations, as illustrated at the top of Figure 8. The two constructions shown (D symmetry) are enantiomorphs whose sense of chirality is related to the way in which the striations are slanted. As n approaches infinity, the symmetry of the constructions approaches Z) in the limit, infinitely many C2 axes are embedded in a plane perpendicular to the C axis. This is the symmetry of a stationary cylinder undergoing a twisting motion, as indicated by the arrows on the cylinders at the bottom of Figure 8, and of an axial tensor of the second rank.41 It is also the helical symmetry of a nonpolar object undergoing a screw displacement, that is, of an object whose enantiomorphism and sense of chirality are T-invariant. 

In summary, objects that exhibit enantiomorphism, whether T-invariant or not, belong to chiral groups. Hence, motion-dependent chirality is encompassed in the group-theoretical equivalent of Kelvin s definition. 

The chirality of objects such as scalene triangles and oriented circles in R2 (Figure 1) and helices in R3 (Figure 3) is a property shared by both enantiomorphs the difference between them is their sense of chirality. In what follows, we shall for simplicity use the term configuration to stand for sense of chirality. Two enantiomorphs are thus said to have opposite configurations. 

It has been recognized147 that chirality measures can be subdivided into two types those that gauge the extent to which a chiroid differs from an achiral reference object (measures of the first kind) and those that gauge the extent to which two enantiomorphs differ from one another (measures of the second kind). In chirality measures of the first kind, the question to be answered is How dissimilar are the chiroid and its achiral reference object In chirality measures of the second kind, the question is How dissimilar are the two enantiomorphs of a chiroid In both cases the underlying concept is that of a distance, measured either between a chiral and an achiral object or between two enantiomorphous chiroids. That is, the degree of chirality of a chiroid X is defined in relation to another, chiral or achiral, reference object Xref The less these two objects match, the more chiral is X. 

A chiral object and its mirror image are enantiomorphous, and they are each other s enantiomorphs. Louis Pasteur (Figure 2-37) was the first who suggested that molecules can be chiral. In his famous experiment in 1848, he recrystallized a salt of tartaric acid and obtained two kinds of small crystals which were mirror images of each other as seen by Pasteur s models in Figure 2-38 preserved at Institut Pasteur at Paris. Originally Pasteur may have been motivated to make these large-scale models because Jean Baptiste Biot, the discoverer of optical activity had very poor vision by the time of Pasteur s discovery [42], Pasteur demonstrated chirality to Biot, who was visibly affected... 

Chirality is a concept well known to organic chemists and to all chemists concerned in any way with structure. The geometric property that is responsible for the nonidentity of an object with its mirror image is called chirality. A chiral object may exist in two enantiomorphic forms that are mirror images of one another. Such forms lack inverse symmetry elements, that is, a center, a plane, and an improper axis of symmetry. Objects that possess one or more of these inverse symmetry elements are superimposable on their mirror images they are achiral. All objects belong to one of these categories. 

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