# The Mathematics of the Periodic Table – 07/13/2021 – Marcelo Viana

The periodic table has two elements in the first row, eight in the second and third, 18 in the fourth and fifth, and 32 in the sixth and seventh. I pointed out last week that 2, 8, 18, and 32 are double the perfect squares of 1, 4, 9, 16. Where do these numbers come from? Let’s start with a more familiar problem to explain.

When we shake a loose string, it can move in many different ways. But when the ends are glued, like on a guitar, there are far fewer possible moves before they can be listed.

There is the basic movement in which a single “belly” vibrates up and down. Two bellies that alternately move in opposite directions, stopping the center of the string. Three alternating bellies, separated by two silent points. And so on (the notes on a real guitar are combinations of these movements).

Vibration phenomena such as the movement of the strings are described by the wave equation formulated by Jean D’Alembert (1717–1783) in 1746. The Sturm-Liouville theory, initiated by Jacques Sturm (1803 – 1855) and Joseph Liouville (1809 – 1882) from the 1830s and is still being researched today, shows that solutions under boundary conditions (e.g. that the tips get stuck) with whole Numbers can be listed.

In quantum models of the atom, the movement of electrons around the nucleus is described by another equation, the Schrödinger equation, but much of the mathematical theory still holds true. And the fact that each electron’s orbit has to “close” as it orbits the nucleus is a constraint.

I still remember the fascination of learning how to solve this problem in college. Solutions can be enumerated with three whole numbers: The main number n takes on the values 1, 2, 3…; the angular momentum l ranges from 0 to n-1; and the magnetic moment varies between -l and + l. It follows that for every n there are n² solutions.

As the atomic number (number of electrons) increases, the solutions are filled in increasing order of energy. There is another quantum number, spin, with the values -½ and + ½, which means that each solution can hold up to two electrons.

The chemical properties of an element depend primarily on the structure of its electrons. And so the mathematics of the Schrödinger equation explains the structure of the periodic table with a little more work.

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