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The hidden structure of the periodic system



The hidden structure of the periodic system

A slightly different periodic table: the table of chemical elements, which dates back to Dmitri Mendeleev and Lothar Meyer, is only one example of how objects – in this case chemical elements – can be organized into such a system. The Leipzig researchers illustrate the general structure of a periodic table with this example: The black dots represent the objects ordered by the green arrows. By using an appropriate criterion, objects can be classified into groups (dashed lines) in which the red arrows create a suborder. Credit: Guillermo Restrepo, MPI for Mathematics in Science

The periodic table of the items described by most chemistry books is only one particular case. This tabular overview of chemical elements, which goes back to Dmitri Mendeleev and Lothar Meyer, as well as the approaches of other chemists to organize the elements, involve different forms of representation of a hidden structure of the chemical elements. This is the conclusion drawn by a researcher from the Max Planck Institute for Mathematics and Science in Leipzig and the University of Leipzig in a recent article. The mathematical approach of the scientists of Leipzig is very general and can provide many different periodic systems according to the principle of order and classification – not only for chemistry, but also for many other areas of knowledge .

It is an icon of the natural sciences and appears in most chemistry classrooms: the Periodic Table of Elements, which is celebrating its 150th anniversary this year. The table below is closely related to Dmitri Mendeleev and Lothar Meyer, two researchers who, in the 1860s, created an arrangement of elements based on their atomic masses and similarities. Today, they are sorted by atomic number (which indicates the number of protons in the nucleus of the atom) from light hydrogen (a proton) to synthetic oganesson (118 protons). ). The elements are also classified into groups: The atoms of the same column usually have the same number of electrons in their outer shell.

Periodic table in different variants

At first glance, the periodic table seems to have brought an unambiguous and definitive order to the 118 elements currently known. But appearances can be deceptive, because many points remain controversial: the scientists do not agree on the elements which belong exactly to the third group, below the scandium and the yttrium. For example, the correct position of lanthanum and actinium is debated. On closer inspection, slightly different variants of the periodic table are found in classrooms, lecture theaters and textbooks.

Guillermo Restrepo and Wilmer Leal of the Max Planck Institute for Mathematics and Science of the University of Leipzig are not surprised. For them, there is no unequivocal provision of the elements; depending on the criterion applied for classification, a different periodic table results. Atoms can be subdivided according to the electronic configuration (ie the number and disposition of their electrons), their chemical behavior, their solubility or their occurrence in geological deposits. It is now widely accepted that chemical elements should be classified according to their atomic number and divided into groups according to their electronic configuration. But even for this periodic table, there are many forms of representation. For example: spiral-shaped with various bulges, pyramid-shaped or as a three-dimensional flower.

A common structure behind periodic tables

Guillermo Restrepo and Wilmer Leal have now systematically examined the ambiguity of the periodic table. This has led to results of considerable importance beyond chemistry. As a result, all forms of representation of chemical elements are based on a common structure, which mathematicians call an ordered hypergraph. The venerable periodic table of Mendeleyev and Meyer thus offers only a representation of the general structure, which Guillermo Restrepo and Wilmer Leal now postulate. New arrangements can also be derived at any time. Guillermo Restrepo therefore compares the order of the chemical elements to a sculpture on which light falls from different directions. "The different areas of projected shadows are periodic tables, which is why there are so many ways to create these tables, and in a way, the tables of periods are projections." internal structure of the periodic table. "

Scientists in Leipzig are now trying to determine the hidden mathematical structure on which the known periodic tables of chemistry are based. For the moment, they have defined three conditions to be met in order to draw up a periodic table. First, you need objects to order. For Mendeleev, Meyer and the creators of the other known periodic tables of chemistry, it is about the chemical elements. These objects must be arranged according to properties such as the atomic mass or the atomic number (that is to say the number of protons). Finally, a criterion is required to group objects into classes. Mendeleev and Meyer used chemical similarity for this.

The hidden structure of the periodic system

Periodic Table of Chemical Bindings: Each of the 94 circles with chemical element symbols represents the bond that the respective element forms with an organic residue. The links are classified according to their polarization. Where there is a direct arrow connection, the order is clear: the hydrogen bonds, for example, are more polarized than the boron, phosphorus and palladium bonds. The same goes for rubidium with respect to cesium, which has particularly low polarized bonds and is therefore at the bottom of the new periodic table. If there is no direct arrow between two elements, they can still be comparable – if there is a chain of arrows between them. For example, oxygen bonds are more polarized than bromine bonds. Bonds represented by the same color have the same binding behavior and belong to one of the 44 classes. Credit: Guillermo Restrepo, MPI for Mathematics in Science

Periodic table of chemical bonds

"If these three conditions are met, periodic tables can also be created for other chemical objects and even for objects that are not part of the chemistry," says Guillermo Restrepo. He and Wilmer Leal show it by examining the chemical bonds between atoms of 94 elements and different conjugates. The polarizability of 94 monocovalent bonds, where bonds are arranged according to the electronegativity and atomic radius of one of the bonded atoms. For example, fluorine, chlorine or oxygen are highly electronegative and assume relatively low atomic radiation in the compounds. Bonds are then ranked according to their similarity.

"We have studied nearly 5,000 substances composed of two elements in different proportions," explains Guillermo Restrepo. "We then looked for similarities in these data: for example, sodium and lithium are similar because they combine with the same elements in the same proportions (for example with oxygen or chlorine, bromine and of iodine), so we found models that we can use to classify the elements. "

A periodic table as a network instead of a matrix

In the 44 classes of chemical elements, there are some similarities with the main groups of the Mendeleev and Meyer Periodic Table. For example, the alkali metals of sodium and lithium are in a group because they form the same simple salts with halogens such as chlorine or fluorine. Like the elements themselves, the bonds of the four halogens (fluorine, chlorine, bromine and iodine) are also found in the same group. However, some classifications differ considerably from those of the conventional periodic table. For example, carbon and silicon are no longer in the same class because they form very different compounds.

The representation of the periodic table of the chemical bonds has nothing to do either with the familiar arrangement of matrix type of the periodic periodic tables of the elements. Instead, the covalent bonds are represented in a network of circles of different colors. Each circle represents a chemical link and the symbol symbolizes membership in one of the 44 groups. Because now two criteria are used for sorting, there is no clear order of the atoms (as in the tables of Mendeleev and Meyer) – the mathematicians speak of a partial order. The circles are connected to the other circles by one or more arrows, creating an ordered hypergraph.

Periodic tables in other scientific fields

The chemical elements and their compounds can also be represented in completely different periodic tables, depending on the underlying order and the classification principle. In addition: the objects of many other scientific fields and their applications can also be classified in periodic tables. For example, ordered hypergraphs are used in information systems and web browsing. Periodic possible systems also appear when countries are taken into account; these can be classified according to social or economic indicators, as well as geographical proximity or cultural similarity. Other examples can be found in the fields of engineering, environmental science, sociology and many other disciplines. Scientists study not only periodic systems because of their importance for chemistry, but especially because of their applications in many other disciplines.


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More information:
Wilmer Leal et al. Formal structure of the periodic system of elements, Acts of the Royal Society A: Mathematical, Physical and Engineering Sciences (2019). DOI: 10.1098 / rspa.2018.0581

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The hidden structure of the periodic system (June 17, 2019)
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