Quantum Numbers: The Address of the Electron

Written: December 2014

As daunting as quantum numbers may seem, they needn't be. Simply put, they are a relatively simple method of classifying electrons. Many analogies have been used to help explain what quantum numbers are and how they describe the electron (e.g., comparing them to a hotel, describing them as a city, etc.). The problem with this approach is, each analogy becomes overly complex in order to maintain similarity to the actual situations quantum numbers describe. So, no analogies will be presented here.

In essence, quantum numbers are the address of the electron. Every specific combination of values for the four numbers specifies exactly and only one electron, just as every specific combination of cities, streets, and numbers specifies exactly and only one house.

1. Principle Quantum Number (n)

The principle quantum number describes the specific shell the electron is in. All atoms have multiple electron spheres or shells, each progressively further from the nucleus. The larger the number of the shell (n), the further away it is from the nucleus (so the shell n=2 is further than n=1, and so on). Another way of characterizing the principle number is seeing it as a description of the energy level of any particular electron. There exists a force of attraction between the positively charged proton and the negatively charged electron, and it takes energy to resist this force. So, the further away an electron is from the nucleus (composed of protons and neutrons), the more energy it must have. The less energy, the closer it will be. In short, the principle quantum number is a description of the energy of an electron.

2. Orbital Quantum Number (ℓ)

The orbital quantum number describes the specific orbital the electron is in. Every electron shell is composed of orbitals. For this reason, orbitals are also known as subshells. The orbital basically describes the shape of the space the electron is likely to be found in. There are four shapes an electron orbital may assume. In other words, there are four possible values of ℓ: 0, 1, 2, and 3. Higher values of ℓ describe more and more complex shapes. "ℓ=0" is also known as the s orbital, "ℓ=1" is known as the p orbital, "ℓ=2" is known as the d orbital, and "ℓ=3" is known as the f orbital. 

S - Orbital

P - Orbital

D - Orbital

The first shell (n=1) has only one s orbital. The second shell (n=2) has one s orbital and three p orbitals. The third shell (n=3) has one s orbital, three p orbitals, and five d orbitals. The fourth shell has (n=4) has one s orbital, three p orbitals, five d orbitals, and seven f orbitals. Keep in mind that, as one considers higher and higher shell levels, the orbitals themselves become larger and larger.

3. Magnetic Quantum Number (mℓ)

The magnetic quantum number describes the orientation of a particular orbital in space. In other words, an orbital's alignment along a particular axis. Take the p-orbital. It has three possible "versions", one aligned on the x-axis, one aligned on the y-axis, and one aligned on the z-axis. The magnetic quantum number designates which of these three a particular p-orbital is. The s-orbital, on the other hand, has only one version. This is because no matter which way it is rotated, it will always maintain the same relationship to each axis; it will look exactly the same.

But how does one find out how many versions a particular orbital has? The rule is that for any orbital with orbital number ℓ, that orbital has -ℓ to ℓ versions, with the integer values in between. The s-orbital has a value of ℓ=0, and so has only one version: namely, 0. The p-orbital has a value of  ℓ=1, and so has three versions: -1, 0, and 1. The d-orbital has a value of  ℓ=2, and so has five versions: -2,-1, 0, 1, and 2. Finally, the f-orbital has a value of  ℓ=3, and so has seven versions: -3, -2, -1, 0, 1, 2, and 3.

There is one version of the S - Orbital, and three versions of the P - Orbital.

You'll notice that in the picture of the p-orbitals above, the first orbital is labelled "Px". The "x" subscript indicates that it is the version aligned with the x axis. And that's all versions of an orbital are: different alignments of the same basic shape along a particular axis. Most of the versions of an orbital look exactly the same; they are merely angled differently, as the above picture of the p-orbitals indicates.

4. Spin Projection Quantum Number (ms)

The spin projection quantum number designates the spin of an electron. Pauli's exclusion principle says that no two electrons can have the exact same quantum numbers, for (1) then they would just be the same electron, and (2) electrons have a force of repulsion to other electrons, and so cannot occupy the same space. So, if electrons have the same principal, orbital, and magnetic quantum number, they must have a different fourth number, and this is taken to refer to an electron's spin. That spin can be either up or down. Now, any particular version of an orbital can hold, at max, two electrons. So by designating the shell, the orbital, the orientation of that orbital in space, and the spin of an electron, one has designated exactly and only one electron. The address of the electron is complete.

References

1. "Quantum Numbers and Electron Configurations." Quantum Numbers and Electron Configurations. Purdue University, n.d. Web. 25 Nov. 2014. <http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html>.
2. Boudreaux, Kevin A., Mr. "Quantum Numbers, Atomic Orbitals, and Electron Configurations." Quantum Numbers, Atomic Orbitals, and Electron Configurations. Angelo State Universty, n.d. Web. 25 Nov. 2014. <http://www.angelo.edu/faculty/kboudrea/general/quantum_numbers/Quantum_Numbers.htm>.