Aufbau Principle, Pauli’s Exclusion Principle, Hund’s Rule - Atomic Orbitals and their Energy Order, Definitions, Examples, Practice Problems & FAQs

What do you do in your summer holidays?

One of the most common things that most of us do is to fill the water in the cooler.

Imagine there’s a power cut at your place and you are left with only one bucket of water to fill in the coolers. There are two coolers in your house, one cooler needs only a bucket of water to fill completely and another cooler needs 4 buckets of water to completely fill.

Which one will you choose?

Obviously, you will choose the first one as just adding one bucket of a water cooler will operate properly.

Similarly in chemistry electrons always try to occupy the lowest energy levels and then they move to a higher level based on some principles. The distribution of electrons of the atom among the various available subshells is called the electronic configuration of the atom.

Let’s understand those principles of Aufbau’s principle, Pauli’s exclusion principle and Hund’s rule in deciding the electronic configuration of an atom in this concept page

Table of Content

• Atomic Orbitals and their Energy Order
• Aufbau Principle
• Pauli's Exclusion Principle
• Hund’s Rule
• Practice Problems
• Frequently Asked Questions-FAQs

Atomic Orbitals and their Energy Order

1. Atomic orbitals and their Quantum Numbers

Atoms have their protons and neutrons in their central core with electrons distributed around this central nucleus. Each electron has a specific energy. The probability of finding electrons of that specific energy is called orbital and is defined by the Schrodinger equation. The orbitals also have one or more sub orbitals or subshells with degenerate energy with the orbital. Each of the thesis subshells is classified with notation s, p, d, f and so on. Under electromagnetic field, the degeneracy is lifted such that each of these subshells also has energies slightly different from the main orbital energy.

The orbital and the subshells are characterised by quantum numbers, n, l and m, where ‘n’ stands for a principal quantum number, ‘l’ stands for azimuthal quantum number and ‘m’ stands for a magnetic quantum number. Each orbital principal quantum number has values of integral numbers starting from 1. The subshells have quantum numbers ‘l’, which is zero for s-subshell, 1 for p-subshell, 2 for d-subshell and so on. The total number of subshells for each ‘n’ orbit shall be equal to ‘n’.

The subshell again under magnetic field can have different orientations and split into further subshell of ‘m’ values, which is from -m to +m through zero. In all total ‘m’ values will be 2l+1.

1. Energy level diagram for orbitals and subshells.

The arrangement of orbital from the nucleus outward in terms of their energy is depicted in the figure below.

Energies of subshells for single electron species in the absence of electromagnetic field

Single-electron species' energy is solely dependent on their principal quantum number (n).

Order of Energy, $1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < \dots$

Energies of subshells of multi-electron species in the absence of electromagnetic field

The Principal quantum number (n), Azimuthal quantum number determine the energy of single-electron species (l).

The energy of the subshell in the increasing order is s < p < d < f…..

For the same subshell, the energy of the orbital is decided by the lowest ‘ n+ l’ value

For the same ‘n + l’ value lowest ‘n’ value will have the lowest energy

Energy of the subshells increases in the order: ‘n’ < (n + l),

When two subshells have the same (n+l) value, the energy of the subshell with the lower "n" value is lower.

Order of Energy, $1s < 2s < 2p < 3s < 3p < 3d < 4s < 4p < 4d < 4f < \dots$

Aufbau Principle

According to this, electrons are inserted into atomic orbitals in ascending order of orbital energy. The Aufbau principle states that atomic orbitals that are available and have the lowest energy levels are filled before those that have higher energy levels.

Germanic in origin, the word "Aufbau" roughly translates to "construct" or "build-up." Below is a diagram showing the sequence in which atomic orbitals are filled. Here, the terms "n" and "l" denote the principal quantum number and azimuthal quantum number, respectively.

The location of electrons in an atom and the corresponding energy levels can be understood using the Aufbau principle. For instance, the electronic configuration of carbon with six electrons is $1s^{2}2s^{2}2p^2.$.

Pauli's Exclusion Principle

When we say electrons can get accommodated in the orbital or subshell, a natural question for any limit to the number of electrons that can be present in an orbit or subshell. Pauli’s exclusion principle answers this question.

No two electrons in an atom may share the same set of four quantum numbers, according to Pauli's Exclusion Principle. This means that each electron has its own unique four quantum numbers.

Atomic electrons cannot share the same quantum state. The quantum numbers $n, l, m_l$ and M_s must be unique for each electron in an atom. The principle is empirical because Pauli arrived at his result after studying atomic spectra. The electronic configuration of multi-electron species is governed by this rule.

This ultimately means, that, only two electrons are allowed to share a subshell, and they must have opposite spins.

Pauli's Exclusion principle has two rules:

1. Only Two electrons can occupy the same orbital and

2. The two electrons in the same orbital have antiparallel spins or opposite spins.

In addition to electrons, other particles like half-integer spin are also subject to Pauli's Exclusion principle. Wolfgang Pauli first proposed the idea in 1925.

Examples of Pauli’s exclusion principle

As an illustration of Pauli's Exclusion principle, we use a neutral helium atom. In this, the atom is bound to two electrons, which have oppositely polarised outer shell positions.

The 1s subshell has two electrons:$n=1,l=0$ and $m_l=0$

A schematic of a helium atom illustrates that there is one up an electron and one down electron.

Pauli's Exclusion Principle: Its Importance

1. This idea contributes to the explanation of a wide range of physical phenomena.

2. It demonstrates how the elements work together to create chemical connections.

3. With the use of this rule, periodic tables are also described.

2

Hund’s Rule

This rule states that electron pairing in the p, d, and f orbitals in the degenerate state (of same energy) cannot take place until each orbital of a particular subshell has one electron in it or is solely occupied.

This rule explains that:

• Each orbital in a sublevel fills up once before filling up with the second electron.
• In singly-occupied orbitals, all the electrons have the same spin.

For example, a p-subshell is made of three subshells that are degenerate (same energy). When these three p-subshells are filled with electrons, electrons, have to fill all three subshells and have the same spins. The fourth electron onwards only can occupy a subshell having already an electron and having the opposite spin. Before pairing, the electrons hence have to fill all the empty orbitals.

This can be understood from the fact that electrons with the same charge repel each other. If placed together in the same orbital, they should have more energy to overcome the repulsion energy which increases the energy of the atom. But, atoms like to be in their lowest energy and hence the pairing of electrons in the same orbital has to be avoided as much as possible.

Understanding the electronic configuration and its function is made easier by the image above. When two atoms come into contact with one another, their valence shells will first interact. The atom is least stable when its valence shells are not completely filled. The valence electrons play a key role in determining an element's chemical properties. Elements with similar valence numbers exhibit similar chemical properties.

The atomic number of nitrogen is 7 means in neutral atom nitrogen has also 7 electrons

$N_7-1s^{2}2s^{2}2p^3$

During the filling of electrons in subshells first single electrons are assigned to all orbitals then pairing takes place. In this discussion, we’ll learn the significance of these filling.

Practice Problems

Q1. Select correct statement

A. The energy of orbitals for single electron species only depends on principle quantum number
B. Energy of orbitals for multi electron species only depends on principle quantum number
C. Energy of orbitals for single electron species only depends on both principle quantum number and Azimuthal quantum number
D. None of these

Answer: (A)

Solution: Energy of orbitals for single electron species only depends on principle quantum number (n).

Order of energy for single electron species are: 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < …

Q2. According to the Aufbau rule, 4s is preferred for the filling of electrons in comparison to 3d. Predict during cation formation first electron released from (last electron entered in orbitals of 3d) for Cu

($\mathit{C}\mathit{u}-1{\mathit{s}}^2,2{\mathit{s}}^{2}2{\mathit{p}}^6,3{\mathit{s}}^{2}3{\mathit{p}}^{6}4{\mathit{s}}^{1}3{\mathit{d}}^{10}$)

A. 3d
B. 4s
C. 3p
D. All of these

Answer: (B)
Solution: we know, According to Aufbau rule, 4s is preferred for filling of an electron in comparison to 3d because the energy of 4s (n+l=4) is lower than 3d (n+l = 5). But during cation formation first, electrons will ionize from outer shell 4s.

$\mathit{C}{\mathit{u}}^{+}-1{\mathit{s}}^2,2{\mathit{s}}^{2}2{\mathit{p}}^6,3{\mathit{s}}^{2}3{\mathit{p}}^{6}3{\mathit{d}}^{10}$

$\mathit{C}{\mathit{u}}^{+}-1{\mathit{s}}^2,2{\mathit{s}}^{2}2{\mathit{p}}^6,3{\mathit{s}}^{2}3{\mathit{p}}^{6}3{\mathit{d}}^9$

Q3. Let’s assume 1 electron has 3 spin values then how many maximum electrons can be accommodated in the 2nd shell (n =2)

A. 8
B. 2
C. 12
D. 18

Answer: (C)
Solution: according to our existing theory, 1 electron has two spin value than one orbital can have maximum two electrons. If a hypothetical experimental theory suggests that one electron has a 3 spin value means one orbital can accommodate a maximum of 3 electrons.

For n = 2, $l=0\left(2s\right) \& 1 \left(2p\right)$

2s contains one orbitals, no of electrons = 3 $\left(1s^3\right)$

2s contains three orbitals, no of electrons = 9 $\left(2p^9\right)$

Q4. Which of the following orbitals have the highest energy for principle quantum number (n) = 4 for multi-electron species

A. S
B. P
C. D
D. f

Answer: (D)
Solution: n = 4, 4th shell

4th shell contains - 4s, 4p, 4d & 4f

4f has the highest energy among given options.

Frequently Asked Questions-FAQs

Q1. Does the Aufbau principle apply to all elements?
Answer: For almost every tested element, the Aufbau principle holds true. There are only two metals that defy this rule: copper and chromium.

The Aufbau principle states that the electron configuration of chromium, element number 24, should be [Ar]3d4s2. According to actual experimental data, the value is [Ar]3d5s1. Element No. 29 is copper, which should be [Ar]3d92s2 but has been found to be [Ar]3d104s1..

Q2. What are the constraints placed on the Aufbau principles?
Answer: When an atom is ionised, the Aufbau principle cannot be used to predict its electron configuration. In other words, it doesn't specify which electrons should be taken out of an atom to form an ion.

Q3. Is it necessary to start filling any orbitals with an upward arrow?
Answer: No, you can start filling orbitals with upward or downward spin but make sure, no electron pairing takes place in the orbitals in a sub-shell until each orbital is occupied by one electron with parallel spin.

Q4. What is exchange energy and pairing energy?
Answer: Exchange energy is the energy released when two or more electrons with the same spin exchange their positions in the degenerate orbitals of a subshell. The more options for exchanging more the electron’s stability. The number of exchange pairs is maximum in half-filled orbitals, hence it is more stable compared to partially filled orbitals.

Pairing energy refers to the energy released with paired electrons sharing one orbital. The more the paired electrons more the electron’s stability. The number of pairs is maximum in fully filled orbitals, hence it is more stable compared to partially filled orbitals.

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