Paper Summary

Citation: Henze, H. R., & Blair, C. M. (1931). The number of isomeric hydrocarbons of the methane series. Journal of the American Chemical Society, 53(8), 3077–3085. https://doi.org/10.1021/ja01359a034

Publication: Journal of the American Chemical Society (JACS) 1931

What kind of paper is this?

This is a seminal method paper in the field of mathematical chemistry and chemical graph theory. It presents a novel, analytical method for calculating the number of structural isomers of alkanes (hydrocarbons of the methane series, $C_nH_{2n+2}$).

What is the motivation?

The primary motivation was the lack of a generalizable and accurate formula for enumerating alkane isomers. Prior attempts by mathematicians and chemists in the late 19th and early 20th centuries had resulted in formulas that were either empirical, limited in scope, or produced incorrect counts for hydrocarbons with more than 11 carbon atoms. This work aimed to develop a theoretically sound method that could be extended to any number of carbons.

What is the novelty here?

The core novelty is the recursive relationship it establishes. Instead of trying to find a direct function of the number of carbon atoms ($N$), the authors’ method calculates the number of isomers for a given $N$ by relating it to the known number of isomers of alcohols (or, equivalently, alkyl radicals) of smaller carbon contents. This was a conceptual breakthrough. The methodology involves a structural decomposition of the problem:

  1. Hydrocarbons are classified based on their carbon backbone symmetry. They are divided into those that can be bisected into two smaller alkyl radicals and those that cannot.
  2. For each class, combinatorial formulas are derived that depend on the number of isomeric alcohols ($T_k$) where $k < N$. This turns the problem of counting large graphs into a recurrence relation based on the counts of smaller, simpler sub-graphs.

What experiments were performed?

The work is theoretical, so the “experiments” were computational and enumerative:

  1. Derivation of the recursion formulas: The main effort was the mathematical derivation of the set of equations for each structural class of hydrocarbon.
  2. Calculation: They applied their formulas to calculate the number of isomers for alkanes up to N=40, which was far beyond what was previously possible with any accuracy.
  3. Validation by Exhaustive Enumeration: To prove the correctness of their method, the authors manually drew and counted all possible structural formulas for the undecanes (C₁₁), dodecanes (C₁₂), tridecanes (C₁₃), and tetradecanes (C₁₄). This exhaustive, brute-force check confirmed their calculated numbers and corrected long-standing errors in the literature (e.g., finding 1858 isomers for C₁₄, not 1855).

What were the outcomes and conclusions drawn?

  • Outcome: The paper provides a robust, recursion-based algorithm to correctly calculate the number of structural isomers for any alkane. The authors published a table of correct isomer counts up to C₄₀, with the number for C₄₀ reaching over 62 trillion.
  • Conclusion: The authors concluded that there is no simple, direct formula relating the number of isomers to N. The problem’s inherent complexity requires a recursive approach. Their method, validated by exhaustive enumeration, provided the first truly general and accurate solution to this long-standing problem in chemistry.

This foundational work established the mathematical framework that would later evolve into modern chemical graph theory and computational chemistry approaches for molecular enumeration.

Additional Resources

  • Paper at Journal of the American Chemical Society

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