Quantum Numbers

Atomic Structure : Quantum Numbers

1926 - Erwin Schrödinger

Developed a model of the atom which accounted for the wave and particle-like behavior of electrons. Instead of restricting electrons to limited circular orbits, he developed a new idea of orbitals:

Orbitals - regions in space where electrons are most likely to be found

This no longer violated the Heisenberg Uncertainty Principle as the Bohr model did since the exact location of an electron at any moment in time is not known. (Review Bohr model notes.)

Bohr's one-dimensional model required only one coordinate to describe the size of the circular orbit in which an electron could be found. Schrödinger's three-dimensional model requires three coordinates, or quantum numbers, which describe the orbitals where electrons can be found.

Quantum Numbers

Principle quantum number (n) - describes the SIZE of the orbital. Since the distance from of an electron from the nucleus is directly proportional to the energy of the electron (as described in the Bohr model), the principle quantum number is also a measure of the orbital.
Angular quantum number (l) - describes the SHAPE of the orbital.
- The s orbitals are spherical (l = 0).
- The p orbitals are polar (l = 1).
- The d orbitals are clover-leaf shaped (l = 2).

[Image] [Image] [Image]

Magnetic quantum number (m) - describes an orbital's ORIENTATION in space.
- For s orbitals (l = 0), there is only one orientation possible, so m must equal 0.
- For p orbitals (l = 1), there are three possible orientations, so m can be -1, 0, or 1.
  - (see picture below)
- For d orbitals (l = 2), there are five possible orientations, so m can be -2, -1, 0, 1, or 2.

[Image] [Image] [Image]

Spin quantum number (s) - describes the SPIN or direction (clockwise or counter-clockwise) in which an electron spins. If there are two electrons in any one orbital, they will have opposite spins, that is, one will have a + spin and the other will have a - spin. The maximum number of electron in any one orbital is two.

Shells and Subshells

Shell - A group of orbitals with the same principal quantum number

e.g. The 2s and 2p orbitals.
Subshell - A group of orbitals with the same angular quantum number

e.g. The three 2p orbitals: 2p_x, 2p_y, and 2p_z

Rules for Allowable Combinations of Quantum Numbers

The three quantum numbers (n, l, and m) that describe an orbital must be integers.
"n" cannot be zero. "n" = 1, 2, 3, 4...
"l" can be any integer between zero and (n-1).

e.g. If n = 4, l can be 0, 1, 2, or 3.
"m" can be any integer between -l and +l.

e.g. If l = 2, m can be -2, -1, 0, 1, or 2.
"s" is arbitrarily assigned as + or -, but for any one subshell (n, l, m combination),
there can only be one of each.

Graphical Representation of Allowable
Combinations of Quantum Numbers

Shell n	Subshell l	Subshell Notation	Orientation m	Number of Orbitals
1	0	1s	0	1
2	0	2s	0	1
2	1	2p	-1 0 +1	3
3	0	3s	0	1
	1	3p	-1 0 +1	3
	2	3d	-2 -1 0 +1 +2	5
4	0	4s	0	1
	1	4p	-1 0 +1	3
	2	4d	-2 -1 0 +1 +2	5
	3	4f	-3 -2 -1 0 +1 +2 +3	7

Example Quantum Number Problem

List the quantum numbers of all the electrons in a neon atom.

1s {	1 0 0	2p	2 1 -1
2s {	2 0 0		2 1 0
			2 1 1

Next: "Electron Configurations"