Quantum Number L

[Phoebe Chen 4I]

According to the textbook the quantum number L can be found by n-1. However, when I was doing the textbook problems I learned that s-orbitals correspond to L=0, p-orbitals L=1 and so on. The L values didn't correspond to the principle quantum number (n) by just subtracting 1 from it. Can someone please explain this to me? Thank you.

[MaanasO 1A]

Hi Phoebe!

L is not actually (n-1). The quantum number is given by 0 <= l <= (n-1). The range of L tells you how many subshells, or different types of orbitals, can be found within a given shell.

Let's take n = 2. This corresponds to the 2nd period on the Period Table (Lithium to Neon). If n = 2, then l is in between 0 and 1 (2-1 = 1). Since we deal with only whole numbers,
l = 0, 1. Therefore, we have 2 subshells in the second energy level. As we go down the period, we see more subshells pop up, and by noting trends, s-orbitals correspond to L = 0, p-orbitals correspond to L = 1, d-orbitals correspond to L = 2, and so on.

Quick technicality: the textbook mentions this, but it is possible to have subshells beyond f, like g and h. But those are super theoretical and we definitely don't cover them.

Hope that helps!

[Phoebe Chen 4I]

What if n=5? Would that mean l could be 0, 1, 2, 3, or 4? I guess I'm also confused about s corresponding to l=0 and p corresponding to l=1, and how that l is separate from the possible values found above. The n value determines l but the orbital shape also determines l? It seems to me as if there are two separate l's. I hope that made sense.

[MaanasO 1A]

Hi Phoebe!

Yes so for any n, L will range from 0 to (n-1). So if n = 5, then L can be 0, 1, 2, 3, 4. Also, L doesn't determine the orbital shape necessarily.

The quantum number n gives you the energy level. Each energy level will contain certain subshells (these are s, p, d, and f). We are doing a simplified version of Schrodinger's Wave equation, so these numbers are simply assigned to make life easier. In the first energy level, the wave function tells us that electrons can take on ONLY 1 type of model, which is what we call the s-orbital. In that energy level, electrons can ONLY flow around the nucleus in a spherical fashion. We correlate this to L = 0. When you go n = 2, there are 2 subshells that electrons fall into (based on the math models), and those are s and p. As n increases, the number of subshells increases, so L was created to assign numbers that we associate with different subshells. n-1 happened to generate the exact number of subshells any energy level could have.

Hope that helps!