The enumeration of isomeric structures for polychlorinated n-alkanes

The radical chlorination of normal alkanes is a common industrial process. The reaction involves a simple and well-known mechanism which one encounters in most introductory courses in organic chemistry. One of the features of this reaction is that, regardless of the conditions of the synthesis, any alkane with at least three carbons always produces a mixture of isomeric monochlorides. In some situations consecutive chlorinations produce an even greater variety of products, with the reaction mixture consisting of mono, di, and polychloroalkanes (PCA). Clearly, the length of the n-alkane reactant in the chlorination process determines to a large extent the number of isomeric structures possible for any one type of alkylchloride. The question that then arises is how many of these constitutional isomers exist for a straight-chain alkane with a fixed number of identical substituents? This question would appear to constitute a simple combinatorial problem to which the literature of combinatorics or statistics should provide a solution. However, a review of relevant chemical and mathematical literature revealed only complicated generating functions and no simple solutions to this particular type of problem. Therefore, the derivation of an approach determining the number of constitutional isomers associated with the empirical formula CnH2n+2-rClr appears to be needed given the practical interest in the associated industrial process as well as the environmental concern regarding these products. Using simple concepts of sets and combinatorics, this paper develops formulae to determine the number of possible constitutional isomers of any one type of chlorinated n-alkane, as well as a discussion of some rules for obtaining the numbers of stereoisomers associated with these PCA's. The paper considers straight chains of even and odd length as well as for even and odd numbers of substituents. (C) 1999 Elsevier Science Ltd. All rights reserved.