# The problem of distinguishing two equal chemical formulas | Science

Mathematics and chemistry use formulas. A mathematical formula relates constants and variables through an equality; while a chemistry, seeks to determine a compound from the atoms it contains and the proportion in which they appear. The big problem is that different substances can have the same chemical formula, as observed in 1811 Joseph Louis Gay-Lussac. Wow! This formula, then, does not seem too useful. In 1823, the chemist Justus von Liebig showed that silver fulminate and silver cyanate, both formed by molecules that contained a silver atom, a carbon atom, an oxygen atom and a nitrogen atom, had very different properties: in particular , the first was very explosive and the second was not. If their molecules consisted of the same atoms and in the same amount, the difference between them had to necessarily reside in the way in which those atoms were linked in each molecule. Gay-Lussac reported these studies to the chemist Jöns Jacob von Berzelius. In fact, the latter had already observed a similar phenomenon: he had discovered that racemic acid and tartaric acid appeared to have the same empirical formula (G4H6O6) but did not share the same properties.

Berzelius suggested calling them isomers. That is, two chemical compounds are called isomers if they have the same chemical formula (same relative proportions of the atoms that make up their molecules) but with different chemical structures (ways in which their atoms are linked) and, therefore, different properties and configuration.

In 1875, the mathematician Arthur Cayley, a remarkable polymath and interested from a young age in chemistry, learned of the problem of the enumeration of isomers through his friend, the German chemist Carl Schorlemmer. Cayley decided to use his knowledge of graph theory for List all possible configurations of the isomers of the alkanes of the formula CnH2n + 2. To address the problem he used mathematical objects called *graphs*.

A graph is a set formed by *vertices and edges*, that represent binary relations, the possible unions, between the first ones. The valence of a vertex is the number of edges incident to the vertex. In particular, Cayley used a certain type of graph, the so-called *trees*, those in which any two vertices are connected by exactly one path of edges.

It occurred to Cayley that the vertices of the trees could represent the carbon atoms of the alkanes (those of hydrogen could be added later) and it was then a question of listing all the possible configurations, that is, all the possible links between those vertices. His proposal was based on the number of centers (valence vertices greater than 1) of the chemical formula. Cayley showed that these trees always have valence vertices less than or equal to four and could only have one or two centers.

For example, consider two alkane isomers with five carbon atoms, pentane, of formula C5H12. First, 2,2-dimethylpropane (C (CH3) 4), which has four valence 1 and one valence 4 vertices, which is therefore its only center:

On the other hand, 2-methylbutane ((CH3) 2-CH-CH2-CH3), which has three valence 1 and two valence vertices 2; these last two are the centers:

Based on the mathematical properties of this type of tree, Cayley proposed a formula to calculate the number of isomers of the alkanes CnH2n + 2: its expression was exact for n from 1 to 11, but it failed further. A shame.

He also invented an algorithm that allowed to calculate the number of isomers for n (CnH2n + 2) knowing the number of isomers for n-1 (Cn-1H2n).

To show the practical interest of his proposal, Cayley applied his method to the alcohols of formula CnH2n + 1OH, in which a hydrogen atom was substituted by the OH radical. He applied his theory using the carbon atoms and the OH radical as vertices of the tree. Cayley calculated that there should be eight isomers of pentanol, of formula C5H11OH. All were subsequently identified (1-pentanol, 3-methyl-1-butanol, 2-methyl-1-butanol, 2,2-dimethyl-1-propanol, 3-pentanol, 2-pentanol, 3-methyl-2-butanol and 2-methyl-2-butanol), confirming the Cayley method.

Of course, there are other ways to classify chemical isomers (optical, stable, etc.). But *The Online Encyclopedia of Integer Sequences* has followed Cayley's theory to know the number of alkane isomers. It is estimated that the C167H336 alkane possesses more isomers than particles has the universe.

**Marta Macho-Stadler** She is a professor in the Department of Mathematics of the University of the Basque Country and a member of the Gender Commission of the ICMAT.

** Coffee and Theorems** is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between the mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination:**Agate Rudder** (ICMAT).

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