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Mathematics of the periodic system of elements
Meyer and Mendeleev came across with their periodic systems by classifying and ordering the known elements by about 1869. Order and similarity were based on knowledge of chemical compounds, which gathered together constitute the chemical space by 1869. Despite its importance, very little is known about the size and diversity of this space and even less is known about its influence upon Meyer's and Mendeleev's periodic system. Here we show, by analysing 11,484 substances reported in the scientific literature up to 1869 and stored in Reaxys database, that 80% of the space was accounted by 12 elements, oxygen and hydrogen being those with most compounds. We found that the space included more than 2,000 combinations of elements, of which 5%, made of organogenic elements, gathered half of the substances of the space. By exploring the temporal report of compounds containing typical molecular fragments, we found that Meyer's and Mendeleev's available chemical space had a balance of organic, inorganic and organometallic compounds, which was, after 1830, drastically overpopulated by organic substances. The size and diversity of the space show that knowledge of organogenic elements sufficed to have a panoramic idea of the space. We determined similarities among the 60 elements known by 1869 taking into account the resemblance of their combinations and we found that Meyer's and Mendeleev's similarities for the chemical elements agree to a large extent with the similarities allowed by the chemical space.
Similarity studies are important for chemistry and their applications range from the periodic table to the screening of large databases in the searching for new drugs. In this later case, it is assumed that similarity in molecular structure is related to similarity in reactivity. However, we state that structural formulas can be regarded as abstract representations emerging from the analysis of large amounts of data upon chemical reactivity. Hence, chemical formulas such as organic functions are not direct pictures of the atomic constitution of matter, but signs used to represent similarity in the reactivity of a class of substances. Therefore, reactivity, rather than molecular structure, becomes the fundamental feature of chemical substances. As reactivity is important, chemical identity is given by the relations substances establish with each other, giving place to a network of chemical reactions. We explore similarity in the network rather than in molecular structure. By characterising each substance in terms of the related ones, we show how Category Theory helps in this description. Afterwards, we study the similarity among substances using topological spaces, which leads us to concepts such as closure and neighbourhood, which formalise the intuition of things lying somewhere near around. The second focus of the chapter is the exploration of the potential of closure operators, and of topological closures in particular, as more general descriptors of chemical similarity. As we introduce the formalism, we develop a worked example, concerning the analysis of similarity among chemical elements regarding their ability to combine into binary compounds. The results show that several of the trends of chemical elements are found through the current approach. © 2015 Bentham Science Publishers Ltd Published by Elsevier Inc. All rights reserved.
For more than 150 years, the structure of the periodic system of the chemical elements has intensively motivated research in different areas of chemistry and physics. However, there is still no unified picture of what a periodic system is. Herein, based on the relations of order and similarity, we report a formal mathematical structure for the periodic system, which corresponds to an ordered hypergraph. It is shown that the current periodic system of chemical elements is an instance of the general structure. The definition is used to devise a tailored periodic system of polarizability of single covalent bonds, where order relationships are quantified within subsets of similar bonds and among these classes. The generalized periodic system allows envisioning periodic systems in other disciplines of science and humanities.
It has been claimed that relational properties among chemical substances are at the core of chemistry. Here we show that chemical elements and a wealth of their trends can be found by the study of a relational property: the formation of binary compounds. We say that two chemical elements A and B are similar if they form binary compounds AC and BC, C being another chemical element. To allow the richness of chemical combinations, we also included the different stoichiomet-rical ratios for binary compounds. Hence, the more combinations with different chemical elements, and with similar stoichiometry, the more similar two chemical elements are. We studied 4,700 binary compounds by using network theory and point set topology, we obtained well-known chemical families of elements, such as: alkali metals, alkaline earth metals, halogens, lanthanides, actinides, some transi-tion metal groups and chemical patterns like: singularity principle, knight's move, and secondary periodicity. The methodology applied here can be extended to the study of ternary, quaternary and other compounds, as well as other chemical sets where a relational property can be defined.
For more than 150 years the structure of the periodic system of the chemical elements has intensively motivated research in different areas of chemistry and physics. However, there is still no unified picture of what a periodic system is. Herein, based on the relations of order and similarity, we report a formal mathematical structure for the periodic system, which corresponds to an ordered hypergraph. It is shown that the current periodic system of chemical elements is an instance of the general structure. The definition is used to devise a tailored periodic system of polarizability of single covalent bonds, where order relationships are quantified within subsets of similar bonds and among these classes. The generalised periodic system allows envisioning periodic systems in other disciplines of science and humanities.