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.
n
Energy Level
1
2
3
4
1st energy level
2nd energy level
3rd energy level
4th energy level
Shell
No. Subshells = n2
No. electrons =
2n2
1
4
9
16
2
8
18
32
K
L
M
N
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
l
subshell
(orbital
shape)
No. orbitals = 2l + 1
0
1
2
3
s subshell
p subshell
d subshell
f subshell
1 (1 x s orbitals)
3 (3 x p orbitals)
5 (5 x d orbitals)
7 (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 l subshell
0
1
2
3
s
p
d
f
values of ml
possible orbitals
0
s
-1, 0, 1
px, py, pz
-2, -1, 0, 1, 2
dxy, dxz, dyz, dx2-y2, dz2
-3, -2, -1, 0, 1, 2, 3
Complex
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
2
1 (K)
1 (K)
0 (s)
0 (s)
0 (1s)
0 (1s)
-½
+½
3
4
5
6
7
8
9
10
2
2
2
2
2
2
2
2
(L)
(L)
(L)
(L)
(L)
(L)
(L)
(L)
0
0
1
1
1
1
1
1
(s)
(s)
(p)
(p)
(p)
(p)
(p)
(p)
0 (2s)
0 (2s)
-1 (2px)
0 (2py)
+1 (2pz)
-1 (2px)
0 (2py)
+1 (2pz)
-½
+½
-½
+½
-½
+½
-½
+½
11
12
13
14
15
16
17
18
3
3
3
3
3
3
3
3
(M)
(M)
(M)
(M)
(M)
(M)
(M)
(M)
0
0
1
1
1
1
1
1
(s)
(s)
(p)
(p)
(p)
(p)
(p)
(p)
0 (3s)
0 (3s)
-1 (3px)
0 (3py)
1 (3pz)
-1 (3px)
0 (3py)
1 (3pz)
-½
+½
-½
+½
-½
+½
-½
+½