Use Madelung’s rule to predict the electronic configuration

Madelung’s rule is an empirical rule that seeks to predict the filling order of energetic subshells in many-electron atoms . This rule was proposed in 1936 by the German physicist Erwin Madelung and, together with the construction principle or Aufbau principle proposed by Niels Bohr, makes it possible to predict the electronic configuration of the first 20 elements of the periodic table, as well as that of the most of the representative elements and some of the transition elements (d and f blocks).

How does Madelung’s rule work?

According to this rule, the energy level of the sublevels of a many-electron atom is determined by the sum of the first two quantum numbers of each sublevel, namely the main quantum number (n) or energy level, and the secondary quantum number (l) or azimuthal quantum number.

In this way, the sublevel that has the lowest energy level is 1s, since it has n=1 and l=0, therefore n+l=1. The following table shows the values ​​of these two quantum numbers for the different subshells that are filled in the known elements of the periodic table, as well as the value of their sum. It should be remembered that the associated values ​​of l for the different types of sublevels (s, p, d and f) are:

• for s, l = 0;
• for p, l = 1;
• for d, l = 2, and
• for f, l = 3.

The list goes on, but no element in its basal state ever fills these sublevels.

Why does the order follow n+l and not just n?

Despite the fact that for the hydrogen atom, which has a single electron, all the subshells for the same value of n have the same energy, this is not the case for polyelectron atoms. This is because the repulsive interactions between electrons in many-electron atoms perturb the subshells, causing them to have different energies. Madelung’s rule predicts in which order the energies of these perturbed subshells actually lie.

As we can see in the table above, the 3d, 4p and 5s subshells all have the same value of n + l = 5, so they should have lower energy than, for example, the 4d subshell.

But how do we know what the energy order is between the 3d, 4p and 5s subshells?

The answer to this question is also provided by Madelung’s Rule, since it has a second part that states that, for the same sum of n+l, the energy order of the subshells is determined by the principal quantum number . In this way, we know then that the 3d sublevel comes first, followed by the 4p and then the 5s.

The Aufbau principle and the Madelung rule

Madelung’s rule alone does not allow us to construct the electronic configuration of an atom or ion. This rule only indicates the energy order of the energy sublevels of an atom. It is thanks to the Aufbau principle or construction principle that we really know how those energy sublevels are filled.

The construction rule tells us that we can imagine polyelectron atoms as atoms that build one proton and one electron at a time. It also states that as we add electrons and protons to an atom in its ground state, the electrons will move to the lowest energy orbital available.

In short, the construction principle tells us that the different energy sublevels are filled from lower to higher energy, and Madelung’s rule tells us what that order of energy is. Together, the Aufbau principle and Madelung’s rule are summarized in what is known as the rain rule, which is a graphical way of representing the filling order of the atomic subshells of many-electron atoms.

Other rules needed to build an electronic configuration

In addition to the Aufbau principle and Madelung’s rule, Hund’s rule and the Pauli exclusion principle are also needed to construct the electronic configuration of an atom. The first indicates that when filling the orbitals of a sublevel with electrons, they must be placed in such a way as to ensure the maximum multiplicity of spin, placing an electron in each orbital first, and another electron can only be placed when all the orbitals of the sublevel have their first electron.

For its part, the Pauli exclusion principle says that if a second electron is to be placed in the same orbital, they must have opposite spins of +1/2 and -1/2. This last principle limits the number of electrons per orbital to 2 and, therefore, the maximum number of electrons in a subshell corresponds to twice the number of orbitals in it. Thus, only 2 electrons fit in the s sublevels, 6 fit in the p, 10 in the d, and 14 in the f.

Now, Madelung’s rule, together with all the other mentioned principles, implies that the order of filling and the maximum number of electrons that fit in each subshell is determined by:

The first row of this table shows all the subshells in order and as an exponent the maximum number of electrons that can fit in each of them. The second row shows the total number of electrons that can fit to completely fill each subshell. For example, the number 48 appearing below 4d ¹⁰ indicates that to completely fill the 4d subshell and all previous subshells, a total of 48 electrons are required.

This table is particularly useful for writing electronic configurations, since, when having the total number of electrons in an atom or ion, it is only necessary to find the number in the second row that is closest to it below. Thus we will know up to which sublevel the atom is completely filled. The remaining electrons are then added to the next subshell.

Let’s see how this is applied in some examples.

Examples of using Madelung’s rule to predict the electronic configuration of an atom or ion

Electron configuration of rubidium (Rb)

Rubidium is element number 37, so it has 37 electrons. The total number of electrons from the previous table that is closest to it below is 36, corresponding to the 4p sublevel. In other words, rubidium has all the sublevels up to 4p completely filled and the difference between 37 and 36, which is only 1 electron, is located in the next sublevel, which is 5s. Therefore, the electronic configuration of rubidium is:

1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ¹

Electronic configuration of sulfur (S)

Sulfur is element 16 and has 16 electrons. Therefore, it fills all the subshells up to 3s, and the remaining 4 electrons (which come from subtracting 16e ^– – 12e ^– ) are located in the 3p subshell:

1s ² 2s ² 2p ⁶ 3s ² 3p ⁴

Electron configuration of iodine(I)

Iodine has 53 electrons so it fills all the subshells up to 4d (with a total of 48 e ^– ) and the remaining 53 – 48 = 5 e ^– go to the 5p subshell:

1s ² 2s ² 2p ⁶ 3s ² 3p ⁶ 4s ² 3d ¹⁰ 4p ⁶ 5s ² 4d ¹⁰ 5p ⁵

Electron configuration of the chloride anion (Cl ^– )

In the case of ions, we must subtract the electric charge (with everything and its sign) from the number of electrons of the neutral element. For example, in the case of chloride, chlorine has 17 e ^– , so chloride must have 17 – (–1)=18 e ^– . As we can see, this number coincides with having the 3p subshell full:

1s ² 2s ² 2p ⁶ 3s ² 3p ⁶

Electronic configuration of the cation calcium (II) (Ca ²⁺ )

Since the charge on calcium is positive, two electrons will be subtracted from the number of electrons in the neutral atom. In this case, it is the 20th atom, so the number of electrons in the calcium cation is 20 – 2 = 18 e ^– . Therefore, it shares the same electronic configuration as the chloride ion.

1s ² 2s ² 2p ⁶ 3s ² 3p ⁶

References

Britannica Encyclopedia. (2008, July 22). Aufbau principle . Encyclopedia Britannica. https://www.britannica.com/science/Aufbau-principle

Chemicool. (2020). Definition of the Madelung Rule . Chemistry Dictionary. https://www.chemicool.com/definition/madelung-rule.html

Luis, J. (2019, September 28). Exceptions to Madelung’s rule in the electronic configuration of chemical elements . TRIPLELINK. https://triplenlace.com/2013/08/06/exceptions-to-the-madelung-rule-and-the-moeller-diagram-in-the-electronic-configuration-of-the-elements- chemicals-2/

Oxford Reference. (2021). Madelung’s rule . Oxford Reference. https://www.oxfordreference.com/view/10.1093/oi/authority.20110803100124745