Isomers are compounds that are identical in formula but different in structure or spatial arrangement. They occur throughout nature but are of special interest in organic chemistry – the study of carbon compounds – because of the huge variety of economically important organic molecules. Scientists have tried to mathematically derive the number of isomers of straight-chain organic molecules, called alkanes, but have discovered no simple relationships between isomer count and carbon content. However, computer programs that decompose alkane structures into manageable fragments give good results.
TL;DR (Too Long; Didn't Read)
It's mathematically impossible to calculate the number of isomers of alkanes, but computer programs use an algorithm to work it out.
Types of Isomers
The two types of isomers are structural and optical. Structural isomers have different arrangements of atoms or small clusters of atoms, called functional groups. These isomers result from differences in branches of molecules branch arrangement of functional groups. Optical isomers, or stereoisomers, are structurally identical but the geometrical positions of their atoms and functional groups are different. Examples of optical isomers include mirror images and molecules that twist in opposite directions.
Meet the Alkanes
Alkanes are chains of carbon (C) and hydrogen (H) atoms. For every n carbon atoms there are (2n + 2) hydrogen atoms. Alkanes originate principally from natural gas and crude oil. The carbon in alkanes forms chains that bind carbon to four other atoms through either C-C or C-H bonds. Straight (acyclic) alkanes don't form ring structures. The simplest alkane is methane (CH4). Alkanes with four or more carbon atoms can form structural isomers, and those with seven or more carbons can also form optical isomers. Some isomers are "sterically unfavorable," meaning that they are unlikely to form because they require extra energy to remain stable.
Robert Paton and Jonathan Goodman at the University of Cambridge offer a free application, called IsoCount, that computes the number of structural and optical isomers for any acyclic alkane. You simply enter the number of carbons in the alkane and the program figures the structural and optical isomer count, noting how many are sterically unfavorable. The program uses an algorithm that iteratively examines a portion of the alkane to derive the number of isomers. For example, if you enter seven, the program reports that the C7H16 alkane has nine structural isomers and two optical ones.
Alkanes with 16 or 17 carbons are not stable compounds and rapidly dissociate at room temperatures. C17 doesn't exist at all and C16 can only form briefly at very low temperatures. Some longer-chain alkanes are also unstable. The IsoCount program accounts for unstable carbon fragments when reporting its results. The count of isomers grows rapidly as the number of carbons in the alkane increases. The IsoCount authors estimate that the isomers of an alkane with 167 carbons outnumber all of the particles in the universe.