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# Pauli Exclusion Principle

20th Feb 2020 @ 4 min read

Physical Chemistry

## Statement

The Pauli exclusion principle states no two electrons in an atom can have the same set of the quantum numbers (principal, azimuthal, magnetic, and spin). In other words, every electron in an atom has a unique set of the quantum numbers.

Since the Pauli exclusion principle is applicable to all fermions (a broader class of subatomic particles), we can generalize the above statement: In a quantum system, each fermion cannot take the same quantum state.

The principle was developed by Wolfgang Pauli, an Austrian physicist in 1925. He was also honored the Nobel Prize for his contribution to physics, especially for his exclusion principle.

## Explanation

In an atom, the four quantum numbers completely describe the quantum state of an electron. The principal quantum represents (n) shells, the azimuthal quantum number (l) defines subshells (s, p, d, f…) in a shell, the magnetic quantum number (ml) represents orbitals in a subshell, and finally the spin quantum number (ms) describes the intrinsic spin of an electron. According to the Pauli exclusion principle, each electron must have a different quantum state, i.e., the set {n, l, ml, ms} is never repeated in an atom.

An electron can only have two spins: ms = 12 and ms = −12. For electrons in an orbital, the value of the principal, azimuthal, and magnetic quantum numbers are the same; however, the spin quantum number is different. Thus, the maximum number of electrons in the same orbital is limited by two since there are only two spin quantum numbers and the repetition of the quantum numbers is prohibited by the exclusion principle.

From the above, it is clear that an orbital can have upmost two electrons. If the first has the positive half spin (ms = 12), the second must have the negative half spin (ms = −12). We often represent these positive and negative spins by up and down arrows, particularly in the energy level diagram.

As we can see from the above diagram, all paired electrons in an orbital have opposite spins. It is the violation of the principle to place parallel spins in an orbital.

The Pauli exclusion principle is fundamental to the structure of the atom. It affects the electronic configuration of the atom. In the absence of the principle, we may not see the piling of electrons around its nucleus, and perhaps all electrons would settle to the lowest energy orbital (1s). Let's explain this with the help of the sodium atom.

The sodium atom has 11 electrons, and its electronic configuration is 1s2 2s2 2p6 3s1. Now take the sodium nucleus (Na11+) with zero electrons. We throw an electron toward the nucleus and it enters the lowest energy level, i.e., 1s. It has either a positive or negative spin. But in our case we assume it always takes the positive. Also, {n, l, ml, ms} = {1, 0, 0, 12}. Now we shoot the second electron, and it must take the negative spin. The quantum numbers for the second electron are {1, 0, 0, −12}.

After the second, the third electron is fired. Because both spins are occupied and no space is left for the third, it settles to the next lowest orbital (2s) with the quantum numbers {2, 0, 0, 12}. Similarly, the fourth electron will have the quantum numbers equal to {2, 0, 0, −12}.

The fifth to tenth electrons will fill 2p orbitals. There are three 2p orbitals: 2pz, 2px, and 2py. Each 2p orbital can take upmost two electrons. Finally, the last electron will enter 3s, which remains unpaired.

Orbitals and quantum numbers of electrons in sodium
Orbital n l ml ms No. of e
1s 1 0 0 ±12 2
2s 2 0 0 ±12 2
2pz 2 1 0 ±12 2
2px 2 1 ±1 ±12 2
2py 2 1 ±12 2
3s 3 0 0 +12 1
Total e 11

From the previous table, each set of the quantum numbers is unique; so, there is no repetition. Because of the number of electrons in an orbital is restricted to two, the stacking or piling of electrons is caused.

### Other than atom

The Pauli exclusion principle is also applicable to other multiple-electron systems like conductors, semiconductors.

### Other Fermions

The principle is not confined to electrons. It is also applicable to other fermions. Fermions are subatomic particles that have half old integers spins, e.g., 12, 32, 52… Electrons are one of them; others are protons, neutrons, quarks, neutrinos, and so on. There is another set of particles called bosons, which have integers spins, e.g., 1, 2, 3, 4… And they do not follow the exclusion principle. Bosons include photons, gluons, mesons, and many others.

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