quantum chemistry - Difference between shells, subshells and orbitals

What are the definitions of these three things and how are they related? I've tried looking online but there is no concrete answer online for this question.

Answer

Here's a graphic I use to explain the difference in my general chemistry courses:

• All electrons that have the same value for $n$ (the principle quantum number) are in the same shell


• Within a shell (same $n$), all electrons that share the same $l$ (the angular momentum quantum number, or orbital shape) are in the same sub-shell


• When electrons share the same $n$, $l$, and $m_l$, we say they are in the same orbital (they have the same energy level, shape, and orientation)

So to summarize:

• same $n$ - shell


• same $n$ and $l$ - sub-shell


• same $n$, $l$, and $m_l$ - orbital

Now, in the other answer, there is some discussion about spin-orbitals, meaning that each electron would exist in its own orbital. For practical purposes, you don't need to worry about that - by the time those sorts of distinctions matter to you, there won't be any confusion about what people mean by "shells" and "sub-shells." For you, for now, orbital means "place where up to two electrons can exist," and they will both share the same $n$, $l$, and $m_l$ values, but have opposite spins ($m_s$).