## Atomic Structure - Quantum Numbers

Unless you go on to study physical chemistry, you won't need quantum numbers again. But for some courses, it's important to know what they are, what they represent, and how they depend on each other.

## What they are

How do we keep track of electrons in an atom? In reality, we don't. We assign each electron a quantum number so that none of them are exactly the same. It also helps us to explain some phenomena that otherwise wouldn't have an explanation.

## About them

- n is the
**principal quantum number**, and it describes the shell number (energy level). If you remember Bohr-Rutherford diagrams (like for sodium, at right), n=1 is the first circle (with 2 electrons on it), n=2 is the next circle with 8 electrons on it, and n=3 is the outermost shell with a single electron on it. - ℓ is the
**azimuthal quantum number**or**angular momentum quantum number**. It describes the shape of the orbital that the electron is in. It goes from 0 to n-1. - mℓ is the
**magnetic quantum number**, and it can go from -ℓ to +ℓ. - ms is the
**spin quantum number**, and it is either +1/2 or -1/2

## Why this matters

How many electrons can fit into the third shell of an atom?

If n=3, then ℓ=0 or 1 or 2. EACH of those ℓ values spawns different mℓ values, depending on how big it is. Finally, each of those is given two different ms values, either +1/2 or -1/2. It turns out that there are 18 electrons total in the third shell. Count the paths in the diagram below to see that there are 18 possible combinations of the quantum numbers.

If n=3, then ℓ=0 or 1 or 2. EACH of those ℓ values spawns different mℓ values, depending on how big it is. Finally, each of those is given two different ms values, either +1/2 or -1/2. It turns out that there are 18 electrons total in the third shell. Count the paths in the diagram below to see that there are 18 possible combinations of the quantum numbers.