Enumeration of constitutional isomers of methyl alkanes by means of alkyl biradicals: equivalence of odd and even isomer series of symmetrical methyl alkanes

Abstract

<p>A mathematical–chemical approach using alkyl biradicals has been used to enumerate isomers of alkenes, alkylcyclopropanes, and alkyl triradicals for alkylcyclobutadienes. In this non-recursive enumeration method for the number of constitutional isomers of methyl alkanes (branched-chain alkanes with all branches to be CH₃), a methyl alkane of <em>n</em> carbon atoms is formed from <em>a</em> number of methyldiyl radicals:CH₂, <em>b</em> number of 1,1-ethanediyl radicals:CH–CH₃, <em>c</em> number of 2,2-propanediyl radical:C–(CH3)₂, and 2 methyl radicals ·CH3, where <em>a, b, and c</em> are solutions of a Diophantine equation <em>a</em> + 2<em>b</em> + 3<em>c</em> + 2 = <em>n</em>. This algorithm does not have to use any previous data on alkyl biradicals and alkanes. Intuitively, in a hydrocarbon isomer series, the number of constitutional isomers of a hydrocarbon of <em>n</em> + 1 carbon atoms should be larger than that of having <em>n</em> carbon atoms, except at the beginning of the series. A graphical proof showed that the conjecture is erroneous for symmetrical methyl alkanes. In addition, the graphical proof showed that even and odd isomer series of symmetrical methyl alkanes are equivalent, i.e., having the equal number of isomers and form equivalent pairs with each pair containing the same number of symmetrical isomers. To my knowledge, this characteristic of a hydrocarbon isomer series has not been reported in the literature since Cayley’s publication on the mathematical theory of isomers.</p>