Achirality diagrams

Analogous to the closely related aldol reactions, reactions of achiral a-substituted enolates and imines lead to mixtures of syn- and rwd-diastereomers. The following diagram shows the postulated pericyclic transition states involved in enolate-iminc condensations10. 

FIGURE 14.1 (a) Diagram of an achiral-chiral system in a heart-cut configuration. MP ... 

The implications for films cast from mixtures of enantiomers is that diagrams similar to those obtained for phase changes (i.e., melting point, etc.) versus composition for the bulk surfactant may be obtained if a film property is plotted as a function of composition. In the case of enantiomeric mixtures, these monolayer properties should be symmetric about the racemic mixture, and may help to determine whether the associations in the racemic film are homochiral, heterochiral, or ideal. Monolayers cast from non-enantiomeric chiral surfactant mixtures normally will not exhibit this feature. In addition, a systematic study of binary films cast from a mixture of chiral and achiral surfactants may help to determine the limits for chiral discrimination in monolayers doped with an achiral diluent. However, to our knowledge, there has never been any other systematic investigation of the thermodynamic, rheological and mixing properties of chiral monolayers than those reported below from this laboratory. 

For a given skeleton, the set of chiral Young diagrams (representations) permits us to define a set of numbers which are characteristic of the chiral properties of the skeleton. In this subsection, we consider the properties of these numbers for the case of achiral ligands, <5 =<3 . 

Table 4 lists, for skeletons i—v, the -chirality functions obtained by the first procedure (%i) and the alternative version of the second procedure (xz) of Section IV, together with the representation of SA and of Young diagrams remain. The parameters A are scalar parameters, the x are pseudoscalar. Different parameters may be used for different representations. 

Fig. 24 Schematic diagram of the formation of tubes from two different DX tiles (yellow and blue in (a)), assembling through sticky ends, represented as complementary shapes. The resulting sheets (b), labeled with biotin (black dots), can fold in register, yielding achiral tubes (c), or with shifted bonds, giving rise to helical tubes (d). The TEM images in (e, f), with scale bar 200 nm, show zigzag arrangement of biotin (bright dots), indicative of chiral structures. Adapted from [108]...

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

Fig. 5 a 3D flow diagram for reactions with antagonistic heterodimers RS. Red lines represent fixed lines. Initial amount of achiral substrate is 0.2c, and the remaining Rate coefficients are k0 = 0, k C = 1, VhetC = 10. b Final ee < >, versus its initial value 0 for a given initial ratio ijo/c. As q0/c decreases, the ee amplification increases... 

The existence of a rigidly achiral presentation suffices as proof of the knot s amphicheirality because all chiral presentations can be isotoped to their mirror images by way of the achiral one. The only possible point group for rigidly achiral presentations of prime (but not composite) knots13 is S2 , n- 1,2,. Figure 23 depicts diagrams of such presentations for a selected number of amphicheiral prime... 

While orientation normally imparts topological chirality, as in the case of the Hopf link, this is not always the case. For example, the Borromean link remains amphicheiral even after orientation.115- 119 This is easily demonstrated with reference to the 5), presentation of the nonoriented link [Figure 27(b)] No matter in which direction the three rings are oriented, the resulting diagram remains centrosymmetric and therefore rigidly achiral. 

A molecular structure such as the one shown in diagram XLII is achiral (presence of a plane of symmetry), but when a b and c = d it can exist in two diastereoisomeric forms. When the two largest or remarkable substituents are on the same side of the double bond, the isomer is cis, and it is designated trans in the other case. Ambiguities have been encountered, and it is recommended to designate as (Z) the isomer with the two sequence rule-preferred substituents on the same side of the double bond (usually the cis-form), and as (E) the other isomer (usually the Wwts-form) (XLIII a>H,c>H) ([53], and refs, therein). 

Figure 2.7 Schematic diagram to represent natural numbers and their conjugates as two spirals that meet at infinity. The mirror image of each spiral represents negative numbers. Real and conjugate number spirals are chiral. In projection on orthogonal axes in the complex plane, together they create an achiral interface...

The question of what product is formed is easily answered as we know we get trans addition to each alkene. These products are, of course, racemic (all reagents achiral) and the diagrams show relative stereochemistry only. 

Recent work reveals a much more complex adsorption phase diagram than was first suspected [28]. For the purposes of highlighting the creation of chirality from the adsorption of achiral molecules on achiral surfaces only the (9 0,1 1), (9 0,-1 1) and p(4x2) phases will be discussed [26]. 

Figure 5. (a) Energy relationships between helical polymers from achiral monomers, (b) Relational diagram between helical polymers from chiral monomers. (Modified from ref 58.)... 

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