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Quantum Numbers

Key Concepts

Electrons can be labelled using the subshell and orbital or by using the four quantum numbers:

  • n : principal quantum number

  • l : azimuthal quantum number

  • ml : magnetic quantum number

  • ms : spin quantum number

Principal Quantum Number, n

The principal quantum number, n, is always a positive integer and tells us the energy level or shell that the electron is found in.

The maximum number of subshells permitted for a particular shell is equal to n2.

The maximum number of electrons permitted in a particular shell is equal to 2 x n2.

nEnergy LevelShellNo. Subshells = n2No. electrons = 2n2
11st energy levelK12
22nd energy levelL48
33rd energy levelM918
44th energy levelN1632

Azimuthal Quantum Number, l

The azimuthal quantum number tells us which subshell the electron is found in, and therefore it tells us the shape of the orbital.

l can have values ranging from 0 to n-1.

The number of orbitals permitted for a particular subshell is equal to 2l + 1.

value of nl = n - 1subshell
(orbital shape)
No. orbitals = 2l + 1
10s subshell1 (1 x s orbitals)
21p subshell3 (3 x p orbitals)
32d subshell5 (5 x d orbitals)
43f subshell7 (7 x f orbitals)

Magnetic Quantum Number, ml

The magnetic quantum number, ml, tells us the orientation of an orbital in space.

ml can have values ranging from -l to +l.

It is not always possible to associate a value of ml with a particular orbital.

value of lsubshellvalues of mlpossible orbitals
0s0s
1p-1, 0, 1px, py, pz
2d-2, -1, 0, 1, 2dxy, dxz, dyz, dx2-y2, dz2
3f-3, -2, -1, 0, 1, 2, 3

Spin Quantum Number, ms

The spin quantum number, ms, tells us the spin of the electron.

ms can have a value of +½ or -½.

Example

The argon atom has 18 electrons.

The quantum numbers for each of the 18 electrons is shown below:

electron n (shell) l (subshell) ml (possible orbital) ms
1 1 (K) 0 (s) 0 (1s)
2 1 (K) 0 (s) 0 (1s)

3 2 (L) 0 (s) 0 (2s)
4 2 (L) 0 (s) 0 (2s)
5 2 (L) 1 (p) -1 (2px)
6 2 (L) 1 (p) -1 (2px)
7 2 (L) 1 (p) 0 (2py)
8 2 (L) 1 (p) 0 (2py)
9 2 (L) 1 (p) +1 (2pz)
10 2 (L) 1 (p) +1 (2pz)

11 3 (M) 0 (s) 0 (3s)
12 3 (M) 0 (s) 0 (3s)
13 3 (M) 1 (p) -1 (3px)
14 3 (M) 1 (p) -1 (3px)
15 3 (M) 1 (p) 0 (3py)
16 3 (M) 1 (p) 0 (3py)
17 3 (M) 1 (p) -1 (3pz)
18 3 (M) 1 (p) -1 (3pz)


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