# How to determine number of structural isomers?

I have come across many questions where I'm asked to give the number of possible structural isomers. For example number, structural isomers of hexane is 5, while the number structural isomers of decane is 75.

How can I determine the possible number of structural isomers of a given organic compound?

• Is this for a specific class of organic molecule or any organic molecule? Because the numbers you gave suggest that the number of isomers quickly increases with size of molecule, and that's just for simple hydrocarbons. And do stereoisomers count? I tried to come up with an equation for simple hydrocarbons, but couldn't identify 5 isomers for hexane, it's too late at night. – user137 Sep 10 '14 at 5:32
• As far as I'm aware, there is generally no simple, formulaic way to do this w/r/t molecules of arbitrary complexity and/or size. Various computational approaches exist; search for, e.g., isomer enumeration. – Greg E. Sep 10 '14 at 6:35
• @ashu This only gives you the degree of unsaturation, which is for alkanes always zero. – Martin - マーチン Sep 10 '14 at 7:46
• If you are given a question like this, I think the purpose is to get you to start drawing different structures and realize which ones are equivalent to ones you have already drawn. – jerepierre Sep 10 '14 at 15:28
• I like that the On-line Encyclopedia of Integer Sequences has a sequence with the number of structural isomers for alkanes with $n$ carbon atoms. The references therein may prove useful in your search for techniques to quantify possible isomers for a given formula. – Nicolau Saker Neto Sep 11 '14 at 2:52

It isn't easy but it is an interesting research topic

Determining the number of possible structures for a given range of chemical formulae isn't simple even for saturated hydrocarbons. The number of possible structural isomers rises rapidly with the number of carbons and soon exceeds your ability to enumerate or identify the options by hand. Wikipedia, for example, lists the numbers of isomers and stereoisomers for molecules with up to 120 carbons. But the counts are getting silly even at 10 carbons where there are 75 isomers and 136 stereoisomers.

It has been an interesting research topic in computational chemistry and mathematics. This old paper (pdf), for example, list some formulae for simple hydrocarbons among other simple series. Part of the interest arises because of the relationship to the mathematics of graph theory (it seems that chemistry has inspired some new ideas in this field of mathematics partially because enumerating possible isomers of hydrocarbons is strongly related to drawing certain simple trees which is intuitively obvious if you use the standard chemical convention of drawing just the carbon backbone and ignoring hydrogens).

You can look up the answers on the fascinating mathematics site OEIS (the online encyclopaedia of integer sequences). The sequence for simple hydrocarbons is here.

But the mathematical approach oversimplifies things from the point of view of real-world chemistry. Mathematical trees are idealised abstract objects that ignore real-world chemical constraints like the fact that atoms take up space in three dimensions. This means that some structures that can be drawn cannot exist in the real world because the atoms are too crowded and cannot physically exist without enough strain to cause them to fall apart.

Luckily, computation chemists have also studied this. There is, unfortunately, no obvious shortcut other than trying to create models of the possible structures and testing them to see if they are too strained to exist. The first two isomers that are too crowded are for 16 and 17 carbons and have these structures:

If you have any intuition of the space filling view of these, you should be able to see why they are problematic. A research group at Cambridge University has produced an applet to enumerate the physically possible isomers for a given number of carbons which is available here if your Java settings allow it. The results are discussed in a paper available in the Journal of Chemical Information and Modelling.

• I've ran regression for these wiki data and for long chains number of isomers rises exponentially and it's about 0.001e^x, where x is no of C atoms. – Mithoron Mar 28 '15 at 17:40

(1) As for the number of alkanes ($\ce{C_nH_{2n+2}}$), Table 1, which is extracted from the data reported in S. Fujita, MATCH Commun. Math. Comput. Chem., 57, 299--340 (2007) (access free), shows the comparison between two enumerations based on Polya's theorem and on Fujita's proligand method.

The number of alkanes ($\ce{C_nH_{2n+2}}$) as constitutional isomers (structural isomers) and as steric isomers is calculated by Polya's theorem (G. Polya and R. C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer (1987)). In the process of calculating constitutional isomers, one 2D structure (graph or constitution) is counted just once. In the process of calculating steric isomers, one achiral molecule or each chiral molecule of an enantiomeric pair is counted just once, where achiral molecules and chiral molecules are not differentiated from each other.

On the other hand, the number of alkanes ($\ce{C_nH_{2n+2}}$) as three-dimensional (3D) structural isomers and as steric isomers is calculated by Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Tree-Dimensional Structures, and Chemical Compounds, Unversity of Kragujevac (2013)). In the process of calculating 3D structural isomers, one achiral molecule or one pair of enantiomers is counted just once, where achiral molecules and chiral molecules (enantiomeric pairs) are differentiated from each other.

For more information, see an account article entitled "Numbers of Alkanes and Monosubstituted Alkanes. A Long-Standing Interdisciplinary Problem over 130 Years" ( S. Fujita, Bull. Chem. Soc. Japan, 83, 1--18 (2010), access free). This account article has discussed the difference between graph enumeration (Polya's theorem) and 3D structural enmeration (Fujita's proligand method) during recursive calculation.

It should be emphasized that graph-theoretical enumerations of chemical compounds as constitutional isomers (structural isomers) and as steric isomers (based on asymmetry) should be differentiated from stereochemical enumerations of chemical compounds as 3D structural isomers and as steric isomers (based on chirality). Although steric isomers based on asymmetry (graphs governed by permutation groups) and steric isomers based on chirality (3D structures governed by point groups) give identical enumeration results, they are conceptually different entities. This point of view stems from Fujita's stereoisogram approach, which is described in a recent book (S. Fujita Mathematical Stereochemistry, De Gruyter (2015)).

(2) Enumeration of achiral and chiral alkanes of a given carbon content has been conducted by considering internal branching ( S. Fujita, Bull. Chem. Soc. Jpn., 81, 1423--1453 (2008)). Figure 3 of this report is cited below.

The symbol [q, t, s, p] means the presence of q quaternary carbons, t tertiary carbons, s secondary carbons, and p primary carbons. Alkanes are categorized into centroidal and bicentroidal alkanes, which are the 3D extension of centroidal and bicentroidal trees of graph theory.

As far as I am aware, there is no straight up formula. Obviously there are very general trends, but not precise enough to tell you the number of isomers.

If you need to decide how many isomers something has, I suggest you either do it manually or use a programme like this: http://www-jmg.ch.cam.ac.uk/tools/isomercount/

This is a start towards a partial answer, but I tried to count the isomers of methane through nonane, and only by moving methyl groups around, each "branch" is only 1 CH3. I noticed a pattern in the number of 1 branch, 2 branch, 3 branch, and 4 branch isomers. Don't know how accurate these numbers are or how well this pattern holds up.

N   Straight    1 Branch    2 Branch    3 Branch    4 Branch
1          1           0           0           0           0
2          1           0           0           0           0
3          1           0           0           0           0
4          1           1           0           0           0
5          1           1           1           0           0
6          1           2           2           0           0
7          1           2           4           0           0
8          1           3           6           1           1
9          1           3           9           4           3


We can calculate the number of structural isomers by double bond equivalent i.e by using the formula:

$$1+ \sum_i \frac{n_i (v_i - 2)}{2}$$

where $n$ repesents the number of atoms $i$ of a given valency, and $v_i$ represents these atoms’ valency. Example:

$$\ce{C4H6}: 1+ \sum_i \frac{n_i (v_i - 2)}{2} = 1+ \frac{4(4-2)}{2}+\frac{6(1-2)}{2} \\ = \frac{8-6}{2} +1 = 2$$ Thus the compound may contain two double bonds or a triple bond and there are two possible isomers.

• The question pertains towards alkanes and the formula that you have specified will be helpful (to a small extent) only for cyclic alkanes. This might as well be a comment than an answer. And Welocme to Chem.SE! – Del Pate Mar 28 '15 at 6:06
• This answer is blatantly wrong. It calculates the double bond equivalents but the example used ($\ce{C4H6}$; i.e. butadiene, cyclobutene or butyne) already has five possible isomers if only a straight chain is assumed. $5 \ne 2$. – Jan Nov 18 '15 at 10:54