Moeller’s diagram: what it is, how to use it in chemistry, and examples

A summary of what the Moeller diagram is and how it is used by applying Madelung's rule.

Chemistry can be especially complicated, so any tool that makes it easier for newcomers to learn chemistry is welcome.

One of the most popular methods to become familiar with Madelung's rule and the electronic configuration of atoms is the Moeller diagram, a graphical mnemonic rule that makes it very easy to see in which orbitals the electrons are located.

In the following we are going to discover what the Moeller diagram consists ofThe Moeller diagram, how it is related to Madelung's rule, how it is applied by means of a few solved examples and which chemical elements do not obey this strategy.

What is the Moeller diagram?

The Moeller diagram, also known as the rain method or diagonal rule, is a graphical and mnemonic method for learning the Madelung rule, a technique for learning and writing the electronic configuration of chemical elements. a graphical and mnemonic method for learning the Madelung rule, a technique for learning and writing the electronic configuration of chemical elements..

This diagram is characterized by drawing diagonals through the columns of the orbitals, from top to bottom from right to left. Through the Moeller diagram an order is defined in the filling of the orbitals, which will be defined by three quantum numbers: n, l and ml.

The Moeller diagram works as follows:

Each column corresponds to a different orbital through which the electrons of an atom, subatomic particles that have a negative charge, circulate. The orbitals in question are: s, p, d and f, each with a specific space to hold electrons and therefore different energy levels..

If we trace the diagonals or arrows in the aforementioned direction, we have that the first orbital is 1s. The second arrow starts from the 2s orbital. The third arrow crosses 2p and 3s. The fourth diagonal is 3p and 4s. The fifth diagonal is 3d, 4p and 5s and so on. The Moeller diagram is an introductory technique for those beginning to study in chemistry the electronic configurations of the elements of the periodic table.

Madelung's rule

Since the Moeller diagram is the graphical representation of Madelung's rule (also known as Klechkovsky's rule in some countries) we must first know what it is. According to this rule, the filling of the orbitals of an atom must obey the following two rules:

Madelung's first rule

Orbitals with the smallest 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. On the other hand, the 4s orbital corresponds to n=4 and l=0, therefore n+l=4+0=4. From this it is established that the electrons fill first the 4s orbital before the 3d, because 4s=4 while 3d=5.

Madelung's second rule

If two orbitals have the same n+l value, the electrons will occupy the one with the lower n value first..

For example, the 3d orbital has a value of n+l=5, identical to that of the 4p orbital (4+1=5) but, since the 3d orbital has the smaller value for n it will be filled first than the 4p orbital.

From all these observations and rules, we can arrive at the following order in the filling of the atomic orbitals: 1s 2s 2p 3s 3p 4s 3d 4p. Although this order is fixed, remembering it by memory is complicated, which is why there is the Moeller diagram that graphically represents its order.

Steps to follow when using the Moeller diagram.

As we have commented in the previous section, the Madelung rule uses the formula n+l to establish which orbitals are filled before and from that determine which is the electronic configuration of a given element. However, the Moeller diagram already represents this graphically and simply, so it is enough to follow the columns of the same diagram and draw diagonals to discover in which order the orbitals of each element are filled.

To discover the electronic configuration of an atom and in which orbitals its electrons are located we must first know its atomic number Z. The Z number corresponds to the number of electrons of an atom, as long as this atom is neutral, that is, it is not an ion, neither positive (cation) nor negative (anion).

Thus, knowing Z for a neutral atom we already know how many electrons a neutral atom of that element usually has. With this in mind, we will begin to draw the diagonals in the Moeller diagram. We must keep in mind that each type of orbital has a different capacity to hold electrons.which are:

• s = 2 electrons
• p = 6 electrons
• d = 10 electrons
• f = 14 electrons

It stops at the orbital where the last electron given by Z has been occupied.

Examples of the Moeller diagram

To better understand how the Moeller diagram works, we will now look at a few practical examples of setting up the electron configuration of different elements.

Beryllium

To establish the electronic configuration of a neutral beryllium (Be) atom, we must first look it up in the periodic table, an alkaline earth which is located in the second column and second row of the periodic table.. Its atomic number is 4, therefore Z=4 and it also has 4 electrons.

Taking all this into account, we are going to use the Moeller diagram to see how the 4 electrons of this element are located. We start by making diagonals in the aforementioned direction, from top to bottom and from right to left.

When we are filling orbitals, it is recommended to put the number of electrons that are in each of them as superscript. As 1s is the first orbital and it occupies two electrons, we will write it:

As we still have free electrons left, we continue filling orbitals. The next one is the 2s orbital and, as with 1s, it occupies 2 electrons, therefore 2s2.therefore 2s2. As we already have all the electrons well placed in the orbitals of the neutral atom of Be we can say that the electronic configuration of this element is:

We make sure that we have done it right by adding the superscripts: 2+2=4

Phosphorus

The element phosphorus (P) is a nonmetal that is found in the third row and column 16 of the periodic tablewith Z=15, so it has 15 electrons in total to occupy the orbitals.

Having seen the previous example, we can move forward a little and place 4 of its electrons in the same orbitals that beryllium has for its 4 electrons, missing 9 more electrons.

After the 2s orbital, the next diagonal enters through the 2p orbital and ends in the 3s orbital. The 2p orbital can occupy 6 electrons, and in the case of 3s, only 2:

At the moment we have 12 electrons well placed, but we are still missing 3 more. We make another diagonal and this time we enter through the 3p orbital according to the Moeller diagram, an orbital that has space for 6 electrons, but as we only have 3 electrons left this orbital will not be completely occupied, putting as superscript a 3.But as we only have 3 electrons left this orbital is not going to be completely occupied, putting as superscript a 3. So, to finish with the phosphorus, its electronic configuration is the following:

We make sure we have done it right by adding the superscripts: 2+2+6+2+2+3=15.

Zirconium

The element zirconium (Zr) is a transition metal and is found in column 4 and row 5 and has a Z=40. Shortening the path by taking advantage of the previous example, we can place the first 18 electrons.

After the 3p orbital, the next to be filled using the Moeller diagram are the 4s, 3d, 4p and 5s orbitals, with capacity for 2, 10, 6 and 2 electrons respectively.

Completing the first nine orbitals of the diagram adds up to a total of 20 electrons, leaving the remaining 2 electrons that are housed in the next orbital, 4d. Thus, the electron configuration of the neutral zirconium element is:

We make sure we got it right by adding the superscripts: 2+2+6+2+2+6+2+2+10+6+2+2+2=40.

Oxygen

Here we see an example a little more complicated that is the one of oxygen (O). This gas is found in column 16 and row 2 of the periodic table, it is a nonmetal and has atomic number 8.

Up to here, seeing the other examples, we would think that its Z=8, however it is not so simple because this gas is of a special nature, being almost always found in the form of an ion with a charge of -2.

This means that, although a neutral oxygen atom does have 8 electrons as indicated by its atomic number, the truth is that in nature it has more, in this case 10 (8 electrons + 2 electrons or, if you prefer, -8 of electric charge -2).

Thus, in this case, the quantity of electrons that we have to place in the orbitals is not 8 but 10 electronsas if we were placing the electrons of the chemical element neon, which does have Z=10.

Having understood this, we just have to do the same as we have been doing in the previous cases only taking into account that we are working with an ion (anion):

We make sure we have done it right by adding the superscripts: 2+2+6=10.

Calcium

To calcium (Ca) something similar happens to oxygen, only in this case we are talking about a cation, that is, a positively charged ion..

This element is found in column 2 row 4 of the periodic table with an atomic number of 20, however, in nature it is usually presented in the form of an ion with positive charge +2, which means that its electronic charge is 18 (- 20 + 2 = 18; 20 electrons - 2 electrons = 18 electrons).

We make sure we have done it right by adding the superscripts: 2+2+6+2+2+6=18.

Exceptions to Moeller's diagram and Madelung's rule

Although Moeller's diagram is very useful to understand Madelung's rule and to know how the electrons of the different chemical elements are located, the truth is that it is not infallible. There are certain substances whose composition does not obey what we have explained.

Their electronic configurations differ experimentally from those predicted by Madelung's rule for quantum reasons.. Among these elements that do not follow the rules are: chromium (Cr, Z=24), copper (Cu, Z=29), silver (Ag, Z=47), rhodium (Rh, Z=45), cerium (Ce, Z=58), niobium (Nb; Z=41), among others.

Exceptions are very frequent when it comes to filling the d and f orbitals. For example, in the case of chromium, which should have a valence configuration ending in 4s^2 3d^4 according to the Moeller diagram and Madelung's rule, it actually has 4s^1 3d^5. Another strange example is that of silver, which instead of having as its last 5s^2 4d^9 has 5s^1 4d^10.