1. Atomic Orbitals: n, ℓ and mℓ
Counting Orbitals I
Atomic Orbitals: n, ℓ and mℓ
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

2. Counting up Orbitals: In subshells
Number of orbitals
in a subshell
Seven f-orbitals
Five d-orbitals
Three p-orbitals
One s-orbital
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

3. Counting up Orbitals: in shells
Number of orbitals
in a subshell
Number of orbitals
in a shell
Seven f-orbitals +
n = 4, 16 orbitals
Five d-orbitals +
n = 3, 9 orbitals
Three p-orbitals +
n = 2, 4 orbitals
One s-orbital
n =1, 1 orbital
1= 12
Because it’s a square
1+3 = 22
Because it’s a square
1+3+5 = 32
Because it’s a square
= 42
Because it’s a square
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

4. Orbitals in shells: Stacking them up
= 42 16
Seven f-orbitals
Five d-orbitals
Three p-orbitals
One s-orbital
n = 4
1+3+5 = 32 9
Five d-orbitals
Three p-orbitals
One s-orbital
n = 3
1+3 = 22 4
Three p-orbitals
One s-orbital
n = 2
In Hydrogen-like Orbitals
All Orbitals in a shell are
Degenerate,
i.e. same energy
n = 1
1= 12
One s-orbital 1
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

5. This works because of the rules of waves. Electrons follow these.
Labeling ℓ and mℓ
Identify mℓ labels
Label ℓ with
number or letter ℓ 3 2 1 +3 +2 +1
Seven f-orbitals
It is
symmetrical.
beautiful. +2 +1 -1
Five d-orbitals +1 -1 -2
Three p-orbitals -1 -2 -3
One s-orbital
This works because of the rules of waves. Electrons follow these.
(“This is bowling. There are rules.” – John Goodman as Walter Sobchak in The Big Lebowski, 1998.
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer

6. Many electron energies5d 4f 6s
Orbitals in shells:
Many electron energies 5p 4d 5s
One electron with each spin: ms = +1/2 and ms = -1/2 for each orbital
Average distance of electrons in same shell is the same.
Energy increases (stability decreases) with increasing ℓ, i.e. s, p, d, f
If all else fails follow the Periodic Table. 4p 4s 3d 1s 1s 3p 2s 6s 6p 5p 4p 3p 2p 6s 6p 5p 4p 3p 2p 6s 6p 5p 4p 3p 2p 3s 3s 4s 3d 2p 5s 4d 2s 6s 5d 1s 4f 5f
Joseph F. Lomax U.S. Naval Academy Copy permission with this footer