6 December 2023 The rule of diagonals It is a method that helps us determine the electronic configuration of atoms and ions. That is, the way in which the electrons are distributed in the different atomic orbitals. The electronic configuration allows us to know the chemical and physical properties of the elements, as well as their behavior in reactions. Know the importance of the rule of diagonals and its application in the study of the chemical elements of the periodic table.

## What is the rule of diagonals?

To apply the rule of diagonals, we must follow the Aufbau principle. This states that electrons occupy the lower energy orbitals first and then the higher energy ones.

Therefore, this method consists of drawing a series of diagonal lines on a diagram that represents the different energy levels and sublevels of atomic orbitals. These lines indicate the order in which the orbitals are filled with electrons, following the Aufbau principle.

The diagram is constructed by placing the Quantum numbers principals (n) and azimuthals (l) in matrix form, as shown below:

n\l 0 1 2 3

1 1s

2 2s 2p

3 3s 3p 3d

4 4s 4p 4d 4f

5 5s 5p 5d 5f

6 6s 6p 6d

7 7s 7p

Each cell represents a energy sublevel, formed by one or more orbitals of the same type. For example, the 1s subshell has a single orbital, the 2p subshell has three orbitals, and the 3d subshell has five orbitals. Each orbital can hold a maximum of two electrons with opposite spins.

Diagonal lines are drawn using the following pattern:

The first diagonal line starts at sublevel 1s and ends at sublevel 2p. The second diagonal line starts at the 2s sublevel and ends at the 3p sublevel. The third diagonal line starts at sublevel 3s and ends at sublevel 4p. And so on, until reaching the highest sublevel.

The order in which the orbitals are filled is obtained by following the lines diagonals from top to bottom and from left to right. Thus, the order would be as follows:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d,…

To write the electronic configuration of an atom or an ion, we must indicate the number of electrons What is in each sublevel? To do this, we use the sublevel symbol and a superscript. For example, the electronic configuration of the hydrogen atom, which has only one electron, would be 1s1.

## What is its relevance in the periodic table?

Through the rule of diagonals we can understand the structure and the organization of the periodic table. To do this, we will first make a reminder.

The periodic table is divided into seven periods (horizontal rows) and 18 groups (vertical columns). Each period corresponds to an energy level (n) and each group corresponds to a type of orbital (s, p, dof).

Thus, elements in group 1 have their last electron in an s orbital, those in groups 13 to 18 have their last electron in a p orbital. Likewise, those in group 3 to 12 have their last electron in a d orbital and those in the f block have their last electron in an f orbital.

Relation to the periodic table

The diagonal rule it shows us that the order in which the orbitals are filled does not exactly match the order of the periods and groups.

For example, the 4s sublevel is filled before the 3d sublevel, even though they belong to the same period. This is because the 4s subshell has a slightly lower energy than the 3d subshell, and therefore the electrons occupy it first.

This method also helps us to explain the Periodic properties of the elements. That is, those that vary regularly according to the position of the elements in the table.

## Applications of the rule of diagonals in chemistry

In chemistry, the diagonal rule has several Practical applications, both theoretical and experimental. Some of these applications are:

• Determine the electronic configuration of atoms and ions in their fundamental or most stable state. This allows us to know its internal structure and its characteristics.
• predict or explain the chemical behavior of the elements and its compounds. For example, we can know what type of chemical bond they form, what type of reactions they carry out or what physical properties they present.
• Establish relationships between the periodic properties of the elements and their position in the periodic table. It is possible to know, for example, how the size of atoms varies or their tendency to gain or lose electrons, depending on their electronic configuration.
• Identify or classify chemical elements using spectroscopic techniques. These techniques are based on the emission or absorption of electromagnetic radiation by atoms when their electrons change their energy level. Here, the diagonal rule helps us to know which transitions are possible and which wavelengths correspond to each one.

## Diagonal Rule Considerations

It is a very useful tool to determine the electronic configuration of atoms and the ions. But, there are some considerations to keep in mind when using it:

• Applies only to atoms and ions in its ground state or more stable. If they are in an excited or less stable state, their electrons can occupy orbitals with higher energy than normal.
• May fail for some items that present exceptions or anomalies in their electronic configuration. These exceptions are due to the fact that some atoms or ions prefer to have a more symmetrical or energetically stable configuration, even if they do not strictly follow the Aufbau principle. For example, the chromium atom has an electronic configuration of 4s13d5, instead of 4s23d4, because it thus has the 3d subshell half-full.
• The rule is invalid for elements beyond atomic number 104, which are called superheavy or transactinide elements. These elements have a very short life and their properties are not precisely known. In addition, its electrons move at speeds close to that of light, which means that relativistic effects that alter its electronic configuration must be taken into account.

### Examples of the rule of diagonals in chemistry

To illustrate how to apply the rule of diagonals in chemistrywe are going to see some examples of electronic configurations of atoms and ions.

carbon atom

It has an atomic number 6, which means it has 6 protons and 6 electrons. Following the rule of diagonals, its electronic configuration is:

1s22s22p2

It means that you have:

• Two electrons in the 1s subshell
• Two electrons in the 2s subshell
• Two electrons in the 2p subshell.

The 2p subshell has three orbitals, but only two are occupied by one electron each, following Hund’s principle.

Calcium ion (Ca2+)

It has an atomic number of 20, but by losing two electrons it is left with 18. According to this method, its electronic configuration is:

1s22s22p63s23p6

It means that you have:

• Two electrons in the 1s subshell
• Two electrons in the 2s subshell
• Six electrons in the 2p subshell
• Two electrons in the 3s subshell
• Six electrons in the 3p subshell.

The 4s subshell, which had two electrons in the neutral atom, is left empty when the ion is formed.

iron atom

It has an atomic number 26, which means it has 26 protons and 26 electrons. According to the rule of diagonals, its electronic configuration would be:

1s22s22p63s23p64s23d6

It means that you have:

• Two electrons in the 1s subshell
• Two electrons in the 2s subshell
• Six electrons in the 2p subshell
• Two electrons in the 3s subshell
• Six electrons in the 3p subshell
• Two electrons in the 4s subshell
• Six electrons in the 3d subshell

The 3d subshell has five orbitals: four are occupied by two electrons each and one by a single electron, according to the same principle.