# Definition of Quantum Numbers

Candela Rocío Barbisan
Chemical engineer

Quantum numbers are defined as integer values ​​that allow identifying the position of an electron within the atom (in its extra-nuclear zone) and thus being able to identify it.

Quantum numbers are a set of numbers represented by letters that, depending on the position of the electron to which they refer, take on different values ​​within a possible range. Next, we are going to describe each one of them and we will see examples of how they are applied according to the electron that we want to designate.

## Principal quantum number (“n”)

It is closely related to the energy that the electron possesses. The higher “n”, the higher the energy, since this number is related to the size of the orbital. Mathematically, it tells us the period in which the electron is located, and as we know from the electronic configurations of the elements of the Periodic Table, there are physically up to seven energy levels. Therefore, “n” can vary from one to seven depending on the distance at which the electron is located from the atom.

## Secondary or azimuthal quantum number (“ℓ”)

This number makes it possible to identify the energy sublevel that the electron is occupying, so again, the higher the azimuthal quantum number, the higher the energy that the electron possesses. Mathematically, “ℓ” will represent the sublevels “s”, “p”, “d” and “f” that we identify electronic configurations of the elements of the Periodic Table. That is why, it can take values ​​that go from zero to (“n”-1) where “n” is the principal quantum number.

For example, if n=1, then ℓ can only be zero, since it corresponds to the energy sublevel “s”. Whereas, if n=2, ℓ can be both zero and one, since we can be referring to an electron from the “s” sublevel or from the “p” sublevel, respectively. In this way we identify: ℓ=0 for energy sublevel “s”, ℓ=1 for energy sublevel “p”, ℓ=2 for energy sublevel “d” and ℓ=3 for energy sublevel “f”.

It should be noted that, according to the “n”, the energy sublevels “s”, “p”, “d” and “f” can add orbitals and, therefore, contain more electrons. For example, at n=1, ℓ=0 with a single “s” subshell and a single orbital that can hold two electrons. For n=2, ℓ=0 with an “s” sublevel or ℓ=1 with the “p” sublevel that can contain three orbitals and accommodate six electrons.

For n=3, ℓ=0 with an “s” subshell or ℓ=1 with the “p” subshell that can contain three orbitals and accommodate six electrons or ℓ=2 with the “d” subshell that can contain five orbitals and hold ten electrons.

Finally, for n=4, ℓ=0 with a sublevel “s” or ℓ=1 with the sublevel “p” that can contain three orbitals and accommodate six electrons or ℓ=2 with the sublevel “d” that can contain five orbitals and accommodate ten electrons or ℓ=3 with the subshell “f” that can contain seven orbitals and accommodate fourteen electrons.

If we wanted to represent these orbitals in space, their form would be something like the following:

Img: ChemistryGod

## Magnetic quantum number (“m”)

It is related to the orientation of the orbital in space and is related to the number of orbitals each sublevel has. Therefore, the value it takes goes from “-ℓ” to “ℓ”. For example, for ℓ=1, subshell “p” contains up to 3 orbitals, so “m” takes values ​​such as -1, 0 or 1. Similarly, for ℓ=2 subshell “d” contains up to 5 orbitals, so “m” can be worth: -2, -1, 0, 1 or 2. Similarly, it is completed for ℓ=0 or ℓ=4.

## Spin quantum number (“s”)

Related to the magnetic properties of the electron and serve to identify the direction of rotation of the electrons that are located within the same orbital, since each of them will have a different sign. Therefore, “s” can take the value of +1/2 or -1/2.

Let’s take Chlorine as an example, to identify quantum numbers in its electrons housed in the last energy level. For this we need to know its electronic configuration, which is: 1s2 2s2 2p6 3s2 3p5. The electrons in the last level are those housed in level 3, therefore: n=3. Then, ℓ=0 or ℓ=1, for the electrons housed in subshells “s” or “p” respectively.

Now, for ℓ=0 (3s2), m=0 and s is worth +1/2 and -1/2 respectively in each of the electrons housed there. For ℓ=1 (3p5), m=-1,0,1, while s is worth +1/2 and -1/2 respectively in each of the electrons housed there for m=-1 and 0, while the orbital designated as m =1 is not complete with two electrons, so we must choose s= +1/2 or -1/2, whichever is chosen by convention.

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