 Quantum Numbers Chemistry Tutorial

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

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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.1

The principal quantum number is essentially the same as the n of the Bohr model of the atom.

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

If the principal quantum number is 1, n = 1, then the number of permitted energy sublevels (subshells) = 12 = 1
That is, the first energy level (shell) has only one permitted energy sublevel (subshell).

If the principal quantum number is 2, n = 2, then the number of permitted energy sublevels (subshells) = 22 = 4
That is, in the second energy level (shell) 4 sublevels (subshells) are permitted.

The maximum number of electrons permitted in a particular energy level (shell) is equal to 2 × n2.

If the principal quantum number is 1, n = 1, then the maximum number of electrons permitted in this shell = 2 × 12 = 2
That is, a maximum of 2 electrons can occupy the first energy level (shell).

If the principal quantum number is 2, n = 2, then the maximum number of electrons permitted in this shell = 2 × 22 = 2 × 4 = 8
That is, a maximum of 8 electrons can occupy the second energy level (shell).

n Energy Level Shell No. Subshells = n2 No. electrons = 2n2
1 1st energy level K 1 2
2 2nd energy level L 4 8
3 3rd energy level M 9 18
4 4th 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.2

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

If n = 1, l = 1 - 1 = 0

If n = 2, l = 2 - 1 = 1

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

If l = 0, the number of permitted orbitals = 2 × 0 + 1 = 1

If l = 1, the number of permitted orbitals = 2 × 1 + 1 = 3

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

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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.

If l = 0, ml = 0

If l = 1, ml = -1, or, ml = 0, or, ml = +1

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

value of l subshell values of ml possible orbitals
0 s 0 s
1 p -1, 0, 1 px, py, pz
2 d -2, -1, 0, 1, 2 dxy, dxz, dyz, dx2-y2, dz2
3 f -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.

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Worked 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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Footnotes:

1. The principal quantum number specifies the total number of nodes in the charge cloud.

2. The azimuthal quantum number specifies the number of angular nodes, surfaces of zero probability density that pass through the origin (the nucleus).