Structural Differentiation in Asymmetric Reactions 169 



On the other hand, rotations around finite * simple axes through the 

 angles specified by their multiplicity are motions which meet all criteria 

 specified in rule 4. In fact, in the examples given, superposition of 

 identical groups, if this was possible, was achieved by rotation around 

 simple axes. This can be done for any rigid structure meeting this 

 superposition test, since two identical objects which have one point 

 in common n can always be brought into coincidence by means of a 

 rotation. 9 - 10 We have therefore two complementary situations. If a 

 structure has an alternating axis of symmetry, it can be superposed 

 on its mirror image and therefore cannot be resolved into optical 

 antipodes. If a structure has a finite simple axis greater than one, 

 it contains identical substituents which meet the superposition test 

 specified in rule 4 and which therefore cannot be differentiated from 

 each other in any reaction. It is clear, then, that the symmetry 

 elements which prevent the differentiation of identical groups differ in 

 kind from those preventing resolution into enantiomorphs, although 

 both can be found in the same molecule. This dichotomy of symmetry 

 elements is well illustrated by example 7 and fully resolves the paradox 

 presented. * 



It is of interest to trace the reason for the failure of rule 2 in 

 example 6, since this structure possessed a twofold simple axis. The 

 compound contained four identical substituents, and rotation around 

 this axis permitted the superposition of the members of two pairs of 

 identical substituents but not the mutual superposition of all four. 

 Hence, to exclude the possibility of differentiation of all identical 

 groups in any structure, one cannot set a fixed limit for the multiplicity 

 of simple axes but must relate in some manner the required number, 

 multiplicity, and orientation of simple axes to the number and disposi- 

 tion of identical groups. The so-called symmetry number, which equals 

 the number of indistinguishable positions into which a molecule can 

 be turned by simple rigid rotations, 13 appears to be a useful character- 

 istic for the general solution of this problem. This number has been 

 related to the so-called symmetry group, which is determined by the 

 combination of all symmetry elements which are found in a given 

 structure. The symmetry number must equal, at least, the number 

 of identical substituents to prevent their differentiation. Although the 

 relationship between symmetry group and symmetry number is avail- 

 able in tabular form, the operation of this criterion seems certainly 



* Infinite simple axes are found in linear molecules like hydrogen cyanide or 

 acetylene and lie in the direction of their bonds. Rotation around such an axis 

 does not achieve the superposition of different atoms. 



