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Aufbau Principle

18th Feb 2020 @ 8 min read

Physical Chemistry

Aufbau principle

Statement

The aufbau principle states: In the ground state of an atom, atomic orbitals are filled by electrons in the order of their increasing energies. In other words, electrons will occupy the lowest-energy orbital first before filling higher-energy orbitals.

According to the aufbau principle, the order of the filling of orbitals is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p…

The IUPAC defines the aufbau principle as:

A rule for building up the electronic configuration of atoms and molecules. It states that a maximum of two electrons are put into orbitals in the order of increasing orbital energy: the lowest-energy orbitals are filled before electrons are placed in higher-energy orbitals.

Etymology

The non-German reader may think aufbau is the name of a scientist, but it is not. The word aufbau (auf-: up and bau: building) is a german word meaning building up or construction.

the etymology of aufbau
The etymology of aufbau

As the rule says the building up of orbitals takes place with the increasing energy of orbitals. Hence, it is also called as the building-up principle.

Explanation

As the aufbau principle says lower orbitals are filled first before higher orbitals. The order of the filling is presented in the diagram below.

Aufbau principle diagram
Aufbau principle

Each type of orbitals in the above diagram is colored the same and are arranged in the ascending order of the principal quantum number (n) from the top to bottom, for example, 2p, 3p, 4p, 5p, 6p… From the left to right, the orbitals are arranged according to the azimuthal quantum number (l). l = 0 corresponds to the s orbital, l = 1 corresponds to the p orbital, l = 2 corresponds to the d orbital, and l = 3 corresponds to the f orbital.

The order of the filling starts with 1s orbital, which is the smallest and lowest energy orbital. Thus, the first electron enters 1s orbital making the electronic configuration of the hydrogen atom, i.e., 1s1. This is followed by the second electron; it also enters 1s orbital since the s orbital can hold two electrons. This makes the electronic configuration 1s2. It is helium, an inert gas. With every electron that gets added, the atomic number increases by one because a proton also gets added.

After 1s orbital completely filled, the third electron goes into 2s making the electronic configuration 1s2 2s1 of the lithium atom, which is directly below hydrogen in the periodic table. Similarly, we have beryllium, 1s2 2s2.

After 2s, the electron fills 2p—follow the arrow in the diagram. 2p can take six electrons. After 2p, we have 3s, 3p, 4s, and so on. The table below lists the electronic configuration of some elements based on the aufbau principle.

The electronic configuration of elements based on the aufbau principle
ElementSymbolAtomic number (Z)Electronic configuration
HydrogenH11s1
HeliumHe21s2
LithiumLi31s2 2s1
CarbonC61s2 2s2 2p2
OxygenO81s2 2s2 2p4
NeonNe101s2 2s2 2p6
SodiumNa111s2 2s2 2p6 3s1
MagnesiumMg121s2 2s2 2p6 3s2
PhosphorusP151s2 2s2 2p6 3s2 3p3
ArgonAr181s2 2s2 2p6 3s2 3p6
PotassiumK191s2 2s2 2p6 3s2 3p6 4s1
CalciumCa201s2 2s2 2p6 3s2 3p6 4s2
ScandiumSc211s2 2s2 2p6 3s2 3p6 4s2 3d1
IronFe261s2 2s2 2p6 3s2 3p6 4s2 3d6
ZincZn301s2 2s2 2p6 3s2 3p6 4s2 3d10
BromineBr351s2 2s2 2p6 3s2 3p6 4s2 3d10 4p5
KryptonKr361s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6
ZirconiumZr401s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d2
TinSn501s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p2
PlutoniumPu941s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6
6s2 4f14 5d10 6p6 7s2 5f6

A question that may bother readers is from where did this rule originate. Is there any scientific reason behind the principle? Well, there is no such equation that dictates this rule. But there is a scientific logic behind it, and it is n + l rule. The energy of an orbital mainly depends on two quantum numbers: the principal quantum number (n) and azimuthal quantum number (l). The principal quantum number corresponds to the shell while the azimuthal quantum number, subshell. As the value of both quantum numbers increases, the energy of an orbital also increases. One way to express this relation is by n + l rule, better known as the Madelung rule.

The Madelung rule, named after German physicist Erwin Madelung, states:

  1. Electrons in an atom fill the orbital with the lowest value of n + l.
  2. When two or more orbitals have the same value of n + l, the electron will occupy the orbital with the lowest value of n.

The rule is straightforward and detailed in the table below.

Madelung rule or n + l rule
OrbitalPrincipal quantum number, nAzimuthal quantum number, ln + l
1s101
2s202
2p213
3s303
3p314
4s404
3d325
4p415
5s505
4d426
5p516
6s606
6s606
4f437
5d527
6p617
7s707
5f538
6d628
7p718
8s808

From the previous table, the order of orbitals concurs with the order of the aufbau principle in the diagram. The energy of the orbital increases with n + l. Whenever there is a tie in the value, the energy increases with n.

In the diagram, each orbital along the diagonal has the same value of n + l (see below).

The aufbau principle and Madelung rule
The aufbau principle and Madelung rule

Exceptions

The aufbau principle is not a universal rule. Not all atoms obey it, especially transition metals, and lanthanides and actinides. The exceptions are mentioned below.

Transition metals

Around ten transition metals go against the aufbau principle. They are listed in the following table.

Exceptions to the aufbau principle in transition metals
ElementSymbolAtomic numberAufbau's predictionExperimental observed
ChromiumCr24[Ar] 4s2 3d4[Ar] 4s1 3d5
CopperCu29[Ar] 4s2 3d9[Ar] 4s1 3d10
NiobiumNb41[Kr] 5s2 4d3[Kr] 5s1 4d4
MolybdenumMo42[Kr] 5s2 4d4[Kr] 5s1 4d5
RutheniumRu44[Kr] 5s2 4d6[Kr] 5s1 4d7
RhodiumRh45[Kr] 5s2 4d7[Kr] 5s1 4d8
PalladiumPd46[Kr] 5s2 4d8[Kr] 4d10
SilverAg47[Kr] 5s2 4d9[Kr] 5s1 4d10
PlatinumPt78[Xe] 6s2 4f14 5d8[Xe] 6s1 4f14 5d9
GoldAu79[Xe] 6s2 4f14 5d9[Xe] 6s1 4f14 5d10

In each element, the d orbital takes an extra electron from the s orbital, except in palladium where both electrons are consumed by the d orbital.

Lanthanides and actinides

In lanthanides and actinides, ten elements violate the aufbau principle.

Exceptions to the aufbau principle in lanthanides and actinides
ElementSymbolAtomic numberAufbau's predictionExperimentally observed
LanthanumLa57[Xe] 6s2 4f1[Xe] 6s2 5d1
CeriumCe58[Xe] 6s2 4f2[Xe] 6s2 4f1 5d1
GadoliniumGd64[Xe] 6s2 4f8[Xe] 6s2 4f7 5d1
ActiniumAc89[Rn] 7s2 5f1[Rn] 7s2 6d1
ThoriumTh90[Rn] 7s2 5f2[Rn] 7s2 6d2
ProtactiniumPa91[Rn] 7s2 5f3[Rn] 7s2 5f2 6d1
UraniumU92[Rn] 7s2 5f4[Rn] 7s2 5f3 6d1
NeptuniumNp93[Rn] 7s2 5f5[Rn] 7s2 5f4 6d1
CuriumCm96[Rn] 7s2 5f8[Rn] 7s2 5f7 6d1
LawrenciumLr103[Rn] 7s2 5f14 6d1[Rn] 7s2 5f14 7p1

In all the above exceptions, the d orbital takes an electron from the f orbital; thorium and lawrencium are special cases. In thorium, 6d consumes both electrons from 5f while in lawrencium 6d is replaced by 7p.

Inconsistency in the ordering of orbitals

Many elements do not always follow the ordering of orbitals as predicted by the aufbau principle. This is observed in transition metals, lanthanides and actinides. Consider an example of scandium (Z = 21). Its electronic configuration is [Ar] 4s2 3d1 as per the principle, but this contradicts the spectroscopic observation. When the scandium atom is ionized, we presume the electron will be released from the highest energy orbital—3d orbitals as stated by the principle. However, the electron is released from 4s to form Sc+ having the electronic configuration [Ar] 3d1 4s1. This suggests that 3d have more energy than 4s. This behavior is also seen in the remaining transition metals.

A similar trend exists in lanthanides and actinides. For example, the aufbau filling order of neodymium (Z = 60) is [Xe] 6s2 4f4, but the order of leaving is [Xe] 4f4 6s2.

Relativistic effects

For heavier nuclei (Z ≥ 120), the aufbau principle becomes invalid. As the nuclear charge increases, the electrons, particularly nearer to the nucleus, experiences a heavy electrostatic force. The electrons of such nuclei have velocities approaching the speed of light. Thus, we need to account for the relativity theory to the quantum mechanic model.

Why exceptions?

We have already mentioned the exceptions to the rule. But why do we have exceptions? Is there any better rule than this? The answer is no. There is no simple mathematical formula that describes the electronic system. In an atom, we have two electrostatic interactions: the attraction between the positive nucleus and negative electrons, and the repulsion among the negative electrons. Every atom or ion tries to minimize the repulsion and try to reach the lowest energy configuration. And understanding such a system is very complicated, not easy. The explanation of these exceptions is not in the scope of this article.

History

Several scientists had contributed to the development of the aufbau principle and it was developed over the course of time. The notion of the ordering of orbitals was seen in the old quantum mechanics model, which was proposed by Niels Bohr in early the 1920s. At the end of the decade, Charles Janet, a French engineer, introduced the left-step periodic table based on the n + l rule. The concept of the n + l rule was adopted by Erwin Madelung in 1936; he proposed the filling of atomic orbitals with this rule. In 1962, Vsevolod Klechkovsky, a Russian chemist, provided a scientific explanation to the rule using the Thomas–Fermi model.

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