Numbers of Isomers

Table 2 1 presents the number of possible alkane isomers as a function of the num ber of carbon atoms they contain As the table shows the number of isomers increases enormously with the number of carbon atoms and raises two important questions... 

The number of C H2n+2 iso mers has been calculated for values of n from 1 to 400 and the comment made that the number of isomers of C167H336 exceeds the number of particles in the known universe (10 °) These obser vations and the historical background of isomer calcu lation are described in a pa per in the April 1989 issue of the Journal of Chemical Edu cat/on (pp 278-281)... 

Multiple Chiral Centers. The number of stereoisomers increases rapidly with an increase in the number of chiral centers in a molecule. A molecule possessing two chiral atoms should have four optical isomers, that is, four structures consisting of two pairs of enantiomers. However, if a compound has two chiral centers but both centers have the same four substituents attached, the total number of isomers is three rather than four. One isomer of such a compound is not chiral because it is identical with its mirror image it has an internal mirror plane. This is an example of a diaster-eomer. The achiral structure is denoted as a meso compound. Diastereomers have different physical and chemical properties from the optically active enantiomers. Recognition of a plane of symmetry is usually the easiest way to detect a meso compound. The stereoisomers of tartaric acid are examples of compounds with multiple chiral centers (see Fig. 1.14), and one of its isomers is a meso compound. 

The answer to the first question is that you cannot easily calculate the number of isomers. The data in Table 2.1 were determined by a mathematician who concluded that no simple expression can calculate the number of isomers. The best way to ensure that you have written all the isomers of a par ticular- molecular- formula is to work systematically, beginning with the unbranched chain and then shortening it while adding branches one by one. It is essential that you be able to recognize when two different-looking structural formulas are actually the sane molecule written in different ways. The key point is the connectivity of the car bon chain. For example, the following group of structural... 

Cayley s extensive computations have been checked and, where necessary, adjusted. Real progress has been achieved by two American chemists, Henze and Blair Not only did the two authors expand Cayley s computations, but they also improved the method and introduced more classes into the compound. Lunn and Senior , on the other hand, discovered independently of Cayley s problems that certain numbers of isomers are closely related to permutation groups. In the present paper, I will extend Cayley s problems in various ways, expose their relationship with the theory of permutation groups and with certain functional equations, and determine the asymptotic values of the numbers in question. The results are described in the next four chapters. More detailed summaries of these chapters are given below. Some of the results presented here in detail have been outlined before ... 

The fact that the number of isomers of homologous series increases rapidly with the number of C-atoms is -well known. The methods described above allow for more precise statements and for... 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

Referring again to the paper of Lunn and Senior, I conclude these general remarks and turn to the analytic determination of the number of isomers of certain special compounds. 

Geometrical isomerism This is an important topic which played a crucial role in the development of coordination chemistry. Werner used the number of isomers... 

Although four is the maximum possible number of isomers when the compound has two chiral centers (chiral compounds without a chiral carbon, or with one chiral carbon and another type of chiral center, also follow the rules described here), some compounds have fewer. When the three groups on one chiral atom are the same as those on the other, one of the isomers (called a meso form) has a plane of symmetry, and hence is optically inactive, even though it has two chiral carbons. Tartaric acid is a typical case. There are only three isomers of tartaric acid a pair of enantiomers and an inactive meso form. For compounds that have two chiral atoms, meso forms are found only where the four groups on one of the chiral atoms are the same as those on the other chiral atom. 

In most cases with more than two chiral centers, the number of isomers can be calculated from the formula 2", where n is the number of chiral centers, although in some cases the actual number is less than this, owing to meso forms.An interesting case is that of 2,3,4-pentanetriol (or any similar molecule). The middle carbon is not asymmetric when the 2- and 4-carbon atoms are both (/ ) [or both (5)] but is asymmetric when one of them is (/ ) and the other (5). Such a carbon is called a pseudoasymmetric carbon. In these cases, there are four isomers two meso forms and one dl pair. The student should satisfy himself or herself, remembering the rules... 

If there is more than one double bond in a molecule and if W X and Y Z for each, the number of isomers in the most general case is 2", although this number may be decreased if some of the substituents are the same, as in... 

Although rings of four carbons and larger are not generally planar (see p. 177), they will be treated as such in this section, since the correct number of isomers can be determined when this is done " and the principles are easier to visualize (see p. 173). 

Rings with more than two differently substituted carbons can be dealt with on similar principles. In some cases, it is not easy to tell the number of isomers by inspection. The best method for the student is to count the number n of differently substituted carbons (these will usually be asymmetric, but not always, e.g., in 68) and then to draw 2" structures, crossing out those that can be superimposed on others (usually the easiest method is to look for a plane of symmetry). By this means, it can be determined that for 1,2,3-cyclohexanetriol there are two meso compounds and a dl pair and for 1,2,3,4,5,6-hexachlorocyclohexane there are seven meso compounds and a dl pair. The drawing of these structures is left as an exercise for the student. 

The pentacoordinate molecules of trigonal bipyramidal form, like PF5, are a very nice example for the study of the formal properties of stereoisomerizations. They are characterized by an appreciable nonrigidity and they permit the description of kinetics among a reasonable number of isomers, at least in particular cases (see below). Therefore the physical and chemical properties of these molecules have been thoroughly investigated in relation to stereoisomerization. Recent reviews may be found in the literature on some aspects of this problem. Mislow has described the role of Berry pseudorotation on nucleophilic addition-elimination reactions and Muetterties has reviewed the stereochemical consequences of non-rigidity, especially for five- and six-atom families as far as their nmr spectra are concerned. 

Pi the Berry step is seen as a double bending of an equatorial and an apical angle. The two apical ligands become equatorial and two equatorial ones go to apical positions. One of the equatorial ligands, the so-called pivot, is on the fourfold axis of the tetragonal pyramidal intermediate state. The connectivity i.e. the number of isomers reached from a given one in one step, is three. 

The possibility of experimental investigation is, of course, restricted to systems with particular ligand partition, where the number of isomers is not too big. The aim of such investigations is to obtain information about the relative probabilities of the five mechanisms which have been defined, till now, on symmetry grounds only. We think that a comparison of the experimental spectra of relaxation times with the various possible theoretical ones (see Table 3) should furnish answers to that question, provided the assumptions of the model are a good description of the physical system. 

The reaction of Pt(C03)(dppp)] with a modest excess of vicinal diols in CH2C12 solution affords the corresponding [Pt(a,/3-diolato)(dppp)] species under equilibrium conditions, a reaction that is readily reversed by the addition of dry ice to the product. The reaction with triols such as glycerol and alditol carbohydrates also affords the corresponding diolato species, with the reaction exhibiting excellent equilibrium regioselectivities for a number of isomers, of which the 7, 6-threo diols are the most favored. 

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