MOELLER DIAGRAM: WHAT IT CONSISTS OF AND EXERCISES SOLVED - CHEMISTRY

The Moeller diagram or method of the rain is a graphic and mnemonic method to learn the Madelung rule; that is, how to write the electron configuration of an element. It is characterized by drawing diagonals through the columns of the orbitals, and following the direction of the arrow, the appropriate order of the same for an atom is established.

In some parts of the world the Moeller diagram is also known as the rain method. Through this, an order is defined in the filling of the orbitals, which are also defined by the three quantum numbers n, l and ml.

Source: Gabriel Bolívar

A simple Moeller diagram is shown in the image above. Each column corresponds to different orbitals: s, p, d and f, with their respective energy levels. The first arrow indicates that the filling of any atom must begin with the 1s orbital.

Thus, the next arrow must start from the 2s orbital, and then from the 2p through the 3s orbital. In this way, as if it were a rain, the orbitals and the number of electrons they house (4 l +2) are noted.

The Moeller diagram represents an introduction for those who study electron configurations.

What is the Moeller diagram?

Madelung's rule

Since the Moeller diagram consists of a graphical representation of Madelung's rule, it is necessary to know how the latter works. The filling of the orbitals must obey the following two rules:

-The orbitals with the lowest values of n + l are filled first, where n is the principal quantum number, and l is the orbital angular momentum. For example, the 3d orbital corresponds to n = 3 and l = 2, therefore, n + l = 3 + 2 = 5; meanwhile, the 4s orbital corresponds to n = 4 and l = 0, and n + l = 4 + 0 = 4. From the above it is established that the electrons fill the 4s orbital first than the 3d one.

-If two orbitals have the same value of n + l, the electrons will occupy the one with the lowest value of n first. For example, the 3d orbital has a value of n + l = 5, as does the 4p orbital (4 + 1 = 5); but since 3d has the smallest value of n, it will fill first than 4p.

From the two previous observations the following order of filling of the orbitals can be reached: 1s 2s 2p 3s 3p 4s 3d 4p.

Following the same steps for different values of n + l for each orbital, the electronic configurations of other atoms are obtained; which in turn can also be determined by the Moeller diagram graphically.

Steps to follow

Madelung's rule establishes the formula n + l, with which the electron configuration can be “armed”. However, as stated, the Moeller diagram already graphically represents this; so just follow its columns and draw diagonals step by step.

How then do you start the electronic configuration of an atom? To do this, you must first know its atomic number Z, which by definition for a neutral atom is equal to the number of electrons.

Thus, with Z we obtain the number of electrons, and with this in mind we begin to draw diagonals through the Moeller diagram.

The s orbitals can accommodate two electrons (applying the formula 4 l +2), the p six electrons, the d ten, and the f fourteen. It stops at the orbital where the last electron given by Z has been occupied.

For further clarification, below are a series of solved exercises.

Solved exercises

Beryllium

Using the periodic table, the element beryllium is located with a Z = 4; that is, its four electrons must be accommodated in the orbitals.

Starting then with the first arrow in the Moeller diagram, the 1s orbital occupies two electrons: 1s ²; followed by the 2s orbital, with two additional electrons to add 4 in total: 2s ².

Therefore, the electron configuration of beryllium, expressed as 1s ² 2s ². Note that the summation of superscripts is equal to the number of total electrons.

Match

The element phosphorus has a Z = 15, and therefore has 15 electrons in total which must occupy the orbitals. To move forward, you start at once with the 1s ² 2s ² configuration, which contains 4 electrons. Then 9 more electrons would be missing.

After the 2s orbital, the next arrow "enters" the 2p orbital, finally landing in the 3s orbital. As the 2p orbitals can occupy 6 electrons, and the 3s 2 electrons, we have: 1s ² 2s ² 2p ⁶ 3s ².

There are still 3 more electrons missing, which occupy the following 3p orbital according to the Moeller diagram: 1s ² 2s ² 2p ⁶ 3s ² 3p ³, electron configuration of the phosphor.

Zirconium

The element zirconium has a Z = 40. Shortening the path with the 1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ configuration, with 18 electrons (that of the noble gas argon), then 22 more electrons would be missing. After the 3p orbital, the next to fill according to the Moeller diagram are the 4s, 3d, 4p and 5s orbitals.

Filling them completely, that is, 4s ², 3d ¹⁰, 4p ⁶ and 5s ², a total of 20 electrons are added. The 2 remaining electrons are therefore housed in the following orbital: the 4d. Thus, the electron configuration of zirconium is: 1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ² 4d ².

Iridium

Iridium has a Z = 77, so it has 37 additional electrons compared to zirconium. Starting from, that is, 1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ² 4d ¹⁰, we must add 29 electrons with the following orbitals of the Moeller diagram.

Drawing new diagonals, the new orbitals are: 5p, 6s, 4f and 5d. Filling the first three orbitals completely we have: 5p ⁶, 6s ² and 4f ¹⁴, to give a total of 22 electrons.

So 7 electrons are missing, which are in the 5d orbital: 1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ² 4d ¹⁰ 5p ⁶ 6s ² 4f ¹⁴ 5d ⁷.

The above is the electron configuration of iridium,. Note that the 6s ² and 5d ⁷ orbitals are highlighted in bold to indicate that they properly correspond to the valence shell of this metal.

Exceptions to the Moeller diagram and Madelung's rule

There are many elements in the periodic table that do not obey what has just been explained. Their electron configurations differ experimentally from those predicted for quantum reasons.

Among the elements that present these discrepancies are: chromium (Z = 24), copper (Z = 29), silver (Z = 47), rhodium (Z = 45), cerium (Z = 58), niobium (Z = 41) and many more.

Exceptions are very frequent in the filling of the d and f orbitals. For example, chromium should have a valence setting of 4s ² 3d ⁴ according to Moeller's diagram and Madelung's rule, but it is actually 4s ¹ 3d ⁵.

Also, and finally, the valence configuration of silver should be 5s ² 4d ⁹; but it really is 5s ¹ 4d ¹⁰.

References

Gavira J. Vallejo M. (August 6, 2013). Exceptions to Madelung's rule and Moeller's diagram in the electronic configuration of chemical elements. Recovered from: triplenlace.com
My superclass. (sf) What is electron configuration? Recovered from: misuperclase.com
Wikipedia. (2018). Moeller diagram. Recovered from: es.wikipedia.org
Dummies. (2018). How to represent electrons in an energy level diagram. Recovered from: dummies.com
Nave R. (2016). Order of Filling of Electron States. Recovered from: hyperphysics.phy-astr.gsu.edu

MOELLER DIAGRAM: WHAT IT CONSISTS OF AND EXERCISES SOLVED - CHEMISTRY - 2025