Tutorials

# Pauli Exclusion Principle

## Pauli Exclusion Principle Definition

The Pauli Exclusion Principle states that in any atom no electron can have the same four electronic quantum numbers as another electron. Every electron must have different quantum numbers. So, in each electronic orbital (same n, l, and ml) there can be two electrons and they must have different spins. One electron will have ms =+ ½  and the other ms = – ½. Therefore, no two electrons will have the same four quantum numbers.

The spin quantum number (ms) was added to the previously discovered three quantum numbers (n, l, ml) by the Pauli exclusion principle. A positive ms usually indicates spin up and is represented by an upward pointing arrow. A negative ms usually indicates spin down and is represented by a down-facing arrow. The spin quantum number is slightly different from the other quantum numbers because it is not dependent on them. It can only have a value of + ½ or – ½ and these values are independent of all other quantum numbers. The other quantum numbers are all interconnected.

The principle also defines that each orbital can only have two electrons. This definition originates from an orbital being defined by the first three quantum numbers. The remaining quantum spin number only has two possible values. Therefore, according to the definition of the Pauli exclusion principle, the orbital can only hold two electrons.

### Fermions vs. Bosons

This principle applies to all fermions. A fermion is an atomic particle that has a half-integer spin. Commonly known fermions are electrons, protons, and neutrons.  Therefore, all these particles will follow the Pauli exclusion principle.

The alternative to a fermion is a boson. Bosons have integer spins. The most common boson is a photon. There can be many photons in one energy state. In one state, they all have the same quantum number. This is a violation of the Pauli exclusion rule. Since photons are bosons, however, they do not follow the Pauli exclusion rule.

## Applications of the Pauli Exclusion Principle in Chemistry

The Pauli exclusion principle is important when determining the electron shell structure of an atom. It pairs with the Aufbau principle to allow us to know what electron orbitals will be filled. Using the Pauli exclusion principle we know that if there are two electrons in an orbital, one must be spin up (+ ½ ) and one must be spin down (- ½ ) to give them different quantum numbers. However, if there is only one electron in an orbital it can have either a positive or negative spin.

The discovery of the Pauli Exclusion principle also helped to explain some phenomena in the periodic table and the reasons behind how some atoms bond. Particularly for solids, many of the previously unexplained properties were able to be explained using the Pauli exclusion principle.

## Example Problems

#### Helium

The simplest atom to look at is helium. Helium has two electrons in the 1s orbital. The 1s orbital has quantum numbers n =1, l=0, and ml=0. Both electrons will be in this subshell. Therefore, one electron will have quantum numbers n=1, l=0,  ml=0, and ms = +1/2. The other electron will have quantum numbers n=1, l=0, ml=0, and ms=-1/2.

#### Beryllium

Beryllium has four electrons which fill the 1s and 2s orbitals. Below are some examples of electron configurations that would violate the Pauli Exclusion Principle as well as the correct depiction.

All the incorrect options have arrows pointing the same way (indicating the same spin) in the same orbital. This indicates they would have the same four quantum numbers and violate the Pauli exclusion principle.

Next, we can also list out the quantum numbers for each electron to see that no electrons have the same 4 quantum numbers.

We start by filling the 1s shell. That means the principal quantum number n is equal to 1. And the s-orbital is denoted by the value 0 in l.

• Electron 1: n =1, l=0, ml=0, and ms= – ½
• Electron 2: n =1, l=0, ml=0, and ms= + ½

That fills the 1s shell. The next shell is the 2s, which changes the principal quantum number n to 2.

• Electron 3: n =2, l=0, ml=0, and ms= – ½
• Electron 4: n =2, l=0, ml=0, and ms= + ½

Comparing the quantum numbers of all four electrons, none of them are the same. In conclusion, they are following the Pauli exclusion principle.

## History of the Pauli Exclusion Principle

The Pauli exclusion principle was discovered by Wolfgang Pauli in 1925. This principle expanded upon the Bohr model. At the time the principle was first discovered, he applied it only to electrons. Later, the principle expanded to all fermions in 1940 by Pauli.

Wolfgang Pauli received the Nobel prize in physics in 1945 for his discoveries and work in quantum chemistry. He also worked on trying to explain the Zeeman effect and proposed the existence of the neutrino. Pauli was born in Austria in 1900 and died in 1958.