# How do you come up with quantum numbers for a specified electron number? I understand that there are four quantum #'s (n, l, m, s), but I'm having trouble coming up with them for a given electron...

How do you come up with quantum numbers for a specified electron number? I understand that there are four quantum #'s (n, l, m, s), but I'm having trouble coming up with them for a given electron #. Please explain! Thanks! :)

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### 2 Answers

In order to come up with the quantum numbers associated to an electron, you first have to know the electron configuration of the atom.

Electron configuration refers to the distribution of the electrons of an atom in specific orbitals.

For example, the electron configuration of oxygen, which has 8 electrons is:

`1s^2 2s^2 2p^4`

(an image is attached if you're not familiar with electron configurations).

The last electron then, is the fourth electron in the p-orbital (`2p^4`

in the electron configuration given above).

Now, let's discuss quantum numbers.

There are four quantum numbers: Principal Quantum Number, Secondary/Azimuthal Quantum Number, Magnetic Quantum Number and Spin Quantum Number.

Principal quantum number refers to the main energy level of the electron. (Values range from n = 1, 2, ...)

Secondary Quantum Number describes the subshell. (Values are L = 0, 1, ..., n-1)

Magnetic quantum number describes the specific orbital within the subshell (or the orientation). (Values are ml = -L, ..., +L)

Spin quantum number refers to the electron spin. (Values are +1/2 or -1/2).

It is straight forward to determine the 1st and 2nd quantum numbers. The main energy level is the number ("coefficient") in your electron configuration, while the secondary quantum number is related to the "letter". In the case of oxygen (2p^4), the main energy level is 2 so n = 2. L, on the other hand, is 1 (since p corresponds to 1 -- s to 0, d to 2, f to 3, and so on).

The last 2 quantum numbers, on the other hand, can be determined by visualizing the location of the electrons (using box diagram).

In the case of oxygen:

There are 3 orbitals in the p-subshell and oxygen has 4 electrons in that subshell. (Note u - upward pointing arrow for first electron, d - downward pointing arrow for second electron on that orbital)

| ud | u_ | u_ | (_ refers to unoccupied space in orbital; Filled out following Hund's Rule of Multiplicity)

To get ml, we simply label the middle box 0 (then -1, -2, ... going to the left, while +1, +2, ... going to the right.

The last electron is "d" in the "diagram" provided. It is in the first box, which corresponds to an ml = -1. Since it's going down, ms = -1/2 (by convention).

Hence, the associated quantum numbers for the last electron of oxygen are:

n = 2

L = 1

ml = -1

ms = -1/2

You can do the same thing for any other electron in an atom.

**Sources:**

Thanks so much! This helped a lot! But could you explain the box and arrow diagram a bit more with a few examples? I'm still a little confused on that. Thanks again! :)