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

m_{l} : magnetic quantum number

m_{s} : 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 n^{2} .

The maximum number of electrons permitted in a particular shell is equal to 2 x n^{2} .

n Energy Level Shell No. Subshells = n^{2} No. electrons = 2n^{2}
1 1^{st} energy level K 1 2
2 2^{nd} energy level L 4 8
3 3^{rd} energy level M 9 18
4 4^{th} energy level N 16 32

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 n l = n - 1subshell (orbital shape) No. orbitals = 2l + 1
1 0 s subshell 1 (1 x s orbitals)
2 1 p subshell 3 (3 x p orbitals)
3 2 d subshell 5 (5 x d orbitals)
4 3 f subshell 7 (7 x f orbitals)

Magnetic Quantum Number, m_{l}
The magnetic quantum number, m_{l} , tells us the orientation of an orbital in space.
m_{l} can have values ranging from -l to +l .

It is not always possible to associate a value of m_{l} with a particular orbital.

value of l subshell values of m_{l} possible orbitals
0 s 0 s
1 p -1, 0, 1 p_{x} , p_{y} , p_{z}
2 d -2, -1, 0, 1, 2 d_{xy} , d_{xz} , d_{yz} , d_{x2-y2} , d_{z2}
3 f -3, -2, -1, 0, 1, 2, 3

Spin Quantum Number, m_{s}
The spin quantum number, m_{s} , tells us the spin of the electron.
m_{s} 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)
m_{l} (possible orbital)
m_{s}
1
1 (K)
0 (s)
0 (1s)
-½
2
1 (K)
0 (s)
0 (1s)
+½
3
2 (L)
0 (s)
0 (2s)
-½
4
2 (L)
0 (s)
0 (2s)
+½
5
2 (L)
1 (p)
-1 (2p_{x} )
-½
6
2 (L)
1 (p)
-1 (2p_{x} )
+½
7
2 (L)
1 (p)
0 (2p_{y} )
-½
8
2 (L)
1 (p)
0 (2p_{y} )
+½
9
2 (L)
1 (p)
+1 (2p_{z} )
-½
10
2 (L)
1 (p)
+1 (2p_{z} )
+½
11
3 (M)
0 (s)
0 (3s)
-½
12
3 (M)
0 (s)
0 (3s)
+½
13
3 (M)
1 (p)
-1 (3p_{x} )
-½
14
3 (M)
1 (p)
-1 (3p_{x} )
+½
15
3 (M)
1 (p)
0 (3p_{y} )
-½
16
3 (M)
1 (p)
0 (3p_{y} )
+½
17
3 (M)
1 (p)
-1 (3p_{z} )
-½
18
3 (M)
1 (p)
-1 (3p_{z} )
+½

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