Quantum Model
I alluded to the Bohr model being incorrect in the last lesson and it is. With the discovery of particle-wave duality, one of the core ideas of quantum mechanics, the idea of electrons orbiting was quickly discarded. In it is place arose the quantum model where electrons exist in “probability clouds” surrounding the nucleus.
Probability Clouds
The idea of a probability cloud is a bit confusing so lets break down this idea by exploring where it came from.
Heisenberg
German physicist, Werner Heisenberg, determined that we can’t know both an object’s momentum and position at the same time. Basically, we can’t know the motion of a mass, momentum, and at the same time know that masses position. This idea, called the Heisenberg Uncertainty Principle, is the basis for probability clouds.
Since we can determine an electron’s momentum and still get a rough, but not exact idea of its position we are able to predict the region that an electron will exist in. These regions are still called orbitals but they don’t represent fixed orbits around the nucleus. Instead, they define an area surrounding the nucleus where the electron has the highest probability of existing. Hence, the name probability clouds.
Types of Clouds
Not only do these orbitals come in different shapes and sizes, but the electrons that fill up these orbitals can also have different properties. In order, to describe both orbital properties and electron properties in atoms chemists and physicists came up with a standardized notation called quantum numbers. This notation has four parts each of which describes a different property of an electron.
Principal Quantum Number
The first of these is the principal quantum number (n), which describes the energy level of an electron. This idea is nearly identical to Bohr’s idea of orbitals and just like orbitals in Bohr’s model the higher the number the higher an electron’s energy.
As we saw previously in Bohr’s model specific energy levels correspond to specific orbits. We see the exact same thing here and n, which can range from 1 to 7, determines what shapes the electron clouds can have. It doesn’t define the exact shape, but rather puts limits on what shapes are allowed.
We can determine the principal quantum number using the periodic table since each row corresponds to an n value.
Azimuthal Quantum Number
The azimuthal quantum number (l), specifically defines the shape of an electron’s orbital called a sub-shell. There are four sub-shell flavors in total and four corresponding numbers to indicate them. These numbers range from 0 to 3 and only certain l values are allowed depending on the n value of an electron.
Before diving into that idea in more detail let’s define each of the number’s corresponding shells as well as their general shape.
| Azimuthal Number | Sub-Shell Name | Sub-Shell Shape |
| 0 | s | |
| 1 | p | |
| 2 | d | |
| 3 | f |
As I said before only certain l values are allowed and this is ultimately dependent on the n value of an electron. The general rule is that l values can range from 0 up to n-1. For example, if n=4 then l could be equal to 0 up to 3.
We can easily determine the l number of a particular electron by referencing the periodic table. Here we won’t be focused on rows or columns, but blocks. With each block corresponding to a different sub-shell as shown below.
If we go back to our n-1 rule we can see that this corresponds to what we see on the periodic table. There aren’t any p sub-shells with an n=1 nor any f sub-shells with an n=3. The easiest way to remember this is to realize that when you get to a new block you will always start counting with one number higher than where the previous block started. For instance, s starts with 1 so p has to begin with 2, d with 3, and f with 4.
Magnetic Quantum Number
Up next is the magnetic quantum number (ml), which describes the specific orbital or probability cloud that an electron inhabits. These numbers range from -l to +l and describe the orientation of an electrons cloud which we call orbitals. For example, a p sub-shell has three dumbell-shaped orbitals that project in the x, y, and z directions.
An electron could be in any one of these three orbitals and each orbital can hold a maximum of two electrons. Since a p sub-shell has l = 1 its magnetic quantum numbers correspond to -1, 0, 1 and since each orbital can hold two electrons a p-subshell can hold a total 6 possible electrons. It isn’t worth memorizing that -1 corresponds to the p orbital in the z-direction, but instead understanding what these numbers mean overall.
Spin Quantum Number
Lastly, there is the spin quantum number (ms), which describes how an electron is spinning around its axis. Just like planets in the solar system electrons spin about their axis while also orbiting around the nucleus. The ms tells us whether they are spinning clockwise (+1/2) or counter-clockwise (-1/2).
Quantum Numbers Visualized
Since all of these numbers can be a bit abstract let’s take a look at this visually. We will do this in three forms: starting with a view at shells, the periodic table, and the more standard electron configurations encountered in chemistry classes.