Atomic Orbitals — Part 2: Quantum Numbers

Introduction

In my last article (part 1), I went over what atomic orbitals are and how they came about.

Atomic Orbitals

One thing I did not cover was the concept of quantum numbers. Quantum numbers are crucial when it comes to atomic orbitals because it kind of acts as an address, it allows us to identify a specific electron in an orbital.

Each electron is assigned four numbers which allows us to identify the electron, these numbers are known as the quantum numbers of the electron.

There are four quantum numbers:

1. n: The energy level number (Identifies which energy level the electron is in)
2. l: The angular momentum number (Identifies the shape of the orbital)
3. m_L: The magnetic number (Identifies the orbital the electron is in)
4. m_S: The spin number (Identifies whether the electron is spin-up or spin-down)

This might already give you an idea of how we can use this to identify an electron. By knowing n, l, and m_L, we know which two electrons we are talking about in a particular orbital. According to Pauli’s exclusion principle (covered in the last article), electrons can only be placed in an orbital if they are in opposite spins in order to minimise spin repulsions.

Therefore, by knowing the spin number, we’ll know exactly which electron we’re talking about.

Let’s delve into the quantum numbers in more detail

n: The energy level number

Recall that electrons occupy energy levels which are themselves split into subshells.

Hence, if n=1, the electron is at ground state, n=2 talks about an electron in the first excited state and so on.

l: The angular momentum number

The values of l depends on the value of n. l can take on values from 0 to n-1. Hence if n=1, l can only be 0. What does this tell us? Remember that in the first energy level, we only have a 1s subshell which contains only one s orbital. Therefore, just these two numbers tell us that the electron is in the 1s subshell. If we knew what m_s is we’d know if the electron was spin-up or spin-down.

Let’s illustrate this with an example. We know that the He atom has an electron configuration of 1s².

The orbital diagram for He

Notice that the electrons are in opposite spin in order to follow Pauli’s exclusion principle.

Now let’s say that n=1, therefore l=0 (l can only take on values from 0 to 1–1 which is just 0).

Therefore we know that the electron is in the 1s subshell. Let’s say m_s gives us a spin-up number. Therefore, we’re talking about the electron highlighted in blue

The spin up electron in 1s

m_L: The magnetic quantum number

The values of m_L depend on the value of l, it can take on values from -l to l. The value of the magnetic quantum number tells us which orbital it is. For example, if the angular momentum number tells us we are looking at the 2p subshell, the the magnetic quantum number would tell us if we are looking at the p_x, p_y, or p_z orbital.

m_s: The spin quantum number

The values of m_s can only be one of two. Either +1/2 or -1/2. If it’s +1/2, the electron is considered spin-up while if it is -1/2, the electron is considered spin-down.

Putting everything together

These four numbers tell us how to find an electron, so let’s put the four together.

Let’s look at a more complicated energy level, say n=2.

If n=2, l can take on values of either 0 or 1. Therefore, m_L can take on values from -1, 0, or 1.

The values of l tell us that there are two possible subshells. The 0 corresponds to the s subshell while the 1 corresponds to a p subshell. In the event l=0, m_L is also 0, this means that there is only one s orbital in the s subshell. This refers to the 2s orbital (Remember that we are at n=2)

In the event l=1, m_L can be either -1, 0 or 1. -1 corresponds to the p_x orbital, 0 corresponds to the p_y orbital while 1 corresponds to the p_z orbital. Actually, it doesn’t matter which number you assosciate with the orbitals.

This is because all the orbitals in a particular subshell are said to be degenerate (they have the same energy because they are all the same distance away from the nucleus, just oriented along a different axis). Hence, it doesn’t matter which orbital you fill first as it follows Aufbau’s principle.

Now let’s suppose we have Neon which has an electron configuration of 1s²2s²2p⁶. Let’s draw the orbital diagram:

Now let’s say we want to identify the electron with quantum numbers n=2, l=1, m_L = 1 m_s is -1/2.

The electron with the above quantum numbers is the one highlighted in blue.

Let’s look at an even more complicated example with n=3.

If n=3, l can either be 0, 1, or 2

If l is 0, m_L is 0, therefore we talk about the 3s subshell

If l is 1, m_L can either be -1, 0 or 1 which can either be the 3p_x, 3p_y or 3p_z orbital

If l is 2, m_L can either be -2, -1, 0, 1, or 2. Notice that there are 5 possible orbitals, this corresponds to a d subshell.

If we had n=4, l could also be 3 which corresponds to an f subshell.

Let’s try and put everything together now with the following:

n=3, l=2, m_L = -1, m_S = +1/2 for an atom of Cobalt.

Cobalt has the following electron configuration: 1s²2s²2p⁶3s²3p⁶3d⁷4s²

If you draw the orbital diagram for Co, you get this:

n=3 corresponds to the third energy level. l=2 corresponds to the d subshell. m_L corresponds to the second 3d orbital. Thus, m_S = +1/2 corresponds to the following electron:

This is basically just a brief introduction in quantum numbers. Hope you guys enjoyed!