High School Chemistry/Pauli Exclusion Principle

When electrons are found inside an atom, they're restricted to specific areas, or regions within the atom which can be described by orbitals. Let's see what this means in terms of quantum numbers.

Lesson Objectives

• Explain the meaning of the Pauli Exclusion Principle.
• Determine whether or not two electrons can coexist in the same atom based on their quantum numbers.
• State the maximum number of electrons that can be found in any orbital.

No Two Electrons in an Atom Can Have the Same Four Quantum Numbers

How do you know that two electrons are in the same orbital? In order to fully specify an orbital, you need to know the principal quantum number, n, the azimuthal quantum number, ℓ, and the magnetic quantum number, m_l. The values of first three quantum numbers for an electron determine exactly which orbital the electron in. Clearly, then, in order to be in the same orbital, two electrons have to have exactly the same values for n, ℓ, and m_l. Now when two electrons have exactly the same values for n, ℓ, and m_l, they share the same region of space within the atom, and in the last lesson, you learned that that had important consequences in terms of their spins. If you remember back to an earlier section, electrons in the same orbital, sharing the same region of space, had to have different values of m_s. If one electron had m_s = +1/2, then the other had to have m_s = −1/2 and vice versa. Let's take a look at several examples.

Example 1 An electron with n = 2, ℓ = 1, ml = −1, and ms = +1/2 is found in the same atom as a second electron with n = 2, ℓ = 1, and ml = −1. What is the spin quantum number for the second electron? Solution: First electron: n = 1, ℓ = 1, ml = −1, ms = +1/2 Second electron: n = 1, ℓ = 1, ml = −1, ms = ? Since the first three quantum numbers are identical for these two electrons, we know that they are in the same orbital. As a result, the spin quantum number for the second electron cannot be the same as the spin quantum number for the first electron. This means that the spin quantum number for the second electron must be ms = −1/2.

Example 2 .An electron with n = 5, ℓ = 4, ml = 3, and ms = −1/2 is found in the same atom as a second electron with n = 5, ℓ = 4, and ml = 3. What is the spin quantum number for the second electron? Solution: First electron: n = 5, ℓ = 4, ml = 3, ms = −1/2 Second electron: n = 5, ℓ = 4, ml = 3, ms = ? Since the first three quantum numbers are identical for these two electrons, we know that they are in the same orbital. As a result, the spin quantum number for the second electron cannot be the same as the spin quantum number for the first electron. This means that the spin quantum number for the second electron must be ms = +1/2.

Notice that whenever the two electrons' first three quantum numbers are the same, the fourth is different. Let's take a look at a few more examples…

Example 3 Can an electron with n = 1, ℓ = 0, ml = 0, and ms = +1/2 exist in the same atom as a second electron with n = 2, ℓ = 0, ml = 0, and ms = +1/2? Solution: First electron: n = 1, ℓ = 0, ml = 0, ms = +1/2 Second electron: n = 2, ℓ = 0, ml = 0, ms = +1/2 Since these two electrons are in different orbitals, they occupy different regions of space within the atom. As a result, their spin quantum numbers can be the same, and thus these two electrons can exist in the same atom.

Example 4 Can an electron with n = 3, ℓ = 1, ml = −1, and ms = −1/2 exist in the same atom as a second electron with n = 3, ℓ = 2, ml = −1, and ms = −1/2? Solution: First electron: n = 3, ℓ = 1, ml = −1, ms = −1/2 Second electron: n = 3, ℓ = 2, ml = −1, ms = −1/2 Since these two electrons are in different orbitals, they occupy different regions of space within the atom. As a result, their spin quantum numbers can be the same, and thus these two electrons can exist in the same atom.

Example 5 Can an electron with n = 1, ℓ = 0, ml = 0, and ms = +1/2 exist in the same atom as a second electron with n = 2, ℓ = 1, ml = 0, and ms = +1/2? Solution: First electron: n = 1, ℓ = 0, ml = 0, ms = +1/2 Second electron: n = 2, ℓ = 1, ml = 0, ms = +1/2 Since these two electrons are in different orbitals, they occupy different regions of space within the atom. As a result, their spin quantum numbers can be the same, and thus these two electrons can exist in the same atom.

Notice that whenever the two electrons have different values of n, or different values of ℓ, or different values of m_l, they can have the same spin quantum number m_s, because they are not in the same orbital, and thus they are not sharing the same region of space within the atom. Let's take a look at one final example.

Example 6 Can an electron with n = 1, ℓ = 0, ml = 0, and ms = +1/2 exist in the same atom as a second electron with n = 1, ℓ = 0, ml = 0, and ms = +1/2? Solution: First electron: n = 1, ℓ = 0, ml = 0, ms = +1/2 Second electron: n = 1, ℓ = 0, ml = 0, ms = +1/2 Since these two electrons are in the same orbital, they occupy the same region of space within the atom. As a result, their spin quantum numbers cannot be the same, and thus these two electrons cannot exist in the same atom.

Figure 7.2: Wolfgang Pauli, the scientist who first proposed the Pauli Exclusion Principle.

Hopefully after having looked at six different examples, it should be obvious to you that electrons in the same atom with the same spin must be in different orbitals, while electrons in the same orbital of the same atom must have different spins. As a result, no two electrons in the same atom can have exactly the same four quantum numbers. If two electrons have the same n, the same ℓ, and the same m_l, then they are in the same orbital. If they also have the same ms, then they also have the same spin, and that is impossible.

The first scientist to realize that two electrons in the same atom couldn’t have the same four quantum numbers was a man name Wolfgang Pauli (Figure 7.2). In 1925, Pauli stated what has come to be known as the Pauli Exclusion Principle. The Pauli Exclusion Principle states that no two identical fermions (a fancy word for electrons and other subatomic particles like electrons) may occupy the same quantum state in an atom simultaneously. In other words, no two electrons in the same atom can have the same four quantum numbers. If n, ℓ, and m_l are the same, ms must be different such that the electrons have opposite spins.

No Atomic Orbital Can Contain More than Two Electrons

An electron can share its territory, or its orbital, with another electron, but only if the other electron is slightly different – in other words, only if the other electron has a different spin.

There’s a limit to the number of different electrons that can share an orbital, because there's a limit to the number of different spins that those electrons can have. When it comes to spins, though, there are only two possibilities. An electron can either be "spin-up", with m_s = +1/2, or "spin-down", with m_s = −1/2. Therefore, if an orbital has one electron that is "spin-up", and a second electron that is "spin-down", the orbital is full. What if a third electron tried to enter the orbital? Well, if the third electron was "spin-up" it would have trouble sharing the orbital, with the "spin-up" electron that's already there. Similarly, if the third electron was "spin-down", it would have trouble sharing the orbital with the "spin-down" electron that's already there. Since the only two options for the third electron are "spin-up" and "spin-down", there's really nothing that third electron can do – it just has to move on and find a new orbital! To summarize, then, because there are only two possibilities for the spin quantum number of an electron, no atomic orbital can contain more than two electrons.

Lesson Summary

• The Pauli Exclusion Principle states that "no two identical fermions may occupy the same quantum state in an atom simultaneously". That is, no two electrons in an atom can have n, ℓ, m_l, and m_s all the same.
• No atomic orbital can contain more than two electrons.

Review Questions

1. Electrons in the same orbital must have different spin quantum numbers. What is true of the other three quantum numbers for two electrons in the same orbital?
2. Electrons in different orbitals can have the same spin quantum numbers. What is true of the other three quantum numbers for two electrons in different orbitals?
3. Fill in the blank using either the word "can" or "cannot".

   (a) An electron with the quantum numbers n = 1, ℓ = 0, m_l = 0 and m_s = +1/2 _____ exist in the same atom as an electron with the quantum numbers n = 2, ℓ = 0, m_l = 0 and m_s = +1/2

   (b) An electron with the quantum numbers n = 1, ℓ = 0, m_l = 0 and m_s = +1/2 _____ exist in the same atom as an electron with the quantum numbers n = 1, ℓ = 0, m_l = 0 and m_s = −1/2

4. Fill in the blanks using numbers.

   (a) There is only 1 orbital at the n = 1 energy level. Therefore the n = 1 energy level can hold a maximum of __ electrons.

   (b) There are 4 orbitals at the n = 2 energy level. Therefore the n = 2 energy level can hold a maximum of __ electrons.

   (c) There are 9 orbitals at the n = 3 energy level. Therefore the n = 3 energy level can hold a maximum of __ electrons.

   (d) There are 16 orbitals at the n = 4 energy level. Therefore the n = 4 energy level can hold a maximum of __ electrons.

5. What is the maximum number of electrons that can exist in p orbitals at energy levels with n < 3.
6. What is the maximum number of electrons that can exist in p orbitals at energy levels with n < 5.

Vocabulary

Pauli Exclusion Principle

No two fermions may occupy the same quantum state in an atom simultaneously; no two electrons in an atom can have the same four quantum numbers.

This material was adapted from the original CK-12 book that can be found here. This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License