Gchem Lecture 4: Electron Quantum Numbers

Electrons held by an atom can exist only at discrete energy levels--that is electron energy levels are quantized. This quantization is described by a unique "address" for each electron, consisting of four quantum numbers designating the shell, subshell, orbital, and spin.

The First Quantum Number: the "principle" quantum number is the shell number, n. It's related to the size and energy of an orbital. The value of n can be any whole number starting with 1, and generally, the greater the value of n, the greater the electron's energy and average distance from the nucleus.

The Second Quantum number (shape): the subshell number, is denoted by letter L. It describes the shape (and energy) of an electrons orbital. The possible values of L depend on the value of n as follows, L=0, 1, 2,..., n-1. For example, if the principal quantum number is 3, then L could be 0, 1, or 2.
L=0 s subshell
L=1 p subshell
L=2 d subshell
L=3 f subshell

The Third Quantum Number: orbital number, is denoted by m(l). It describes the three-dimensional orientation of an orbital. The possible values of m(l) depend on the value of L as follows: m(l)=-L, -(L-1),..., -1, 0,...(L-1), L. For example, if L=2, the m(l) could be -2, -1, 0, 1, 2
If L=0, then m(l) can only be equal to 0 (one possibility), so each S subshell has just 1 orbital
If L=1, then m(l) can equal -1, 0, 1 (three possibilities), so each P subshell has 3 orbitals
If L=2, then m(l) can equal -2, -1, 0, 1, 2 (five possibilities), so each D subshell has 5 orbitals
If L=3, then m(l) can equal -3, -2, -1, 0, 1, 2, 3 (seven possibilities), so the F subshell has 7 orbitals

The Fourth Quantum Number: spin number, denoted m(s); designates the electron's intrinsic magnetism; spin can either be (+1/2) spin up or (-1/2) spin down. Every orbital can accommodate 2 electrons, one spin up and one spin down. When an orbital is full, we say the electrons are "spin paired".

Here's the alternative/shorthand method to learning this:

Shells-->subshells-->orbitals-->spin number

Shells are the energy levels. The shells have subshells, which in turn have atomic orbitals. The orbitals are within the subshell. The azimuthal quantum numbers s, p, d and f designate the subshells. Consider the p subshell. It has 3 orbitals: px, py and pz. Each orbital can hold a total of 2 electrons, making 6 the total number of electrons a p subshell can house. The d subshell has 5 orbitals, and can hold a total number of ten electrons. The f subshell has 7 orbitals, and can house a total number of 14 electrons. Think of a shell as a neighborhood, a subshell as a house, and the orbitals as the different bedrooms within the house where the electrons stay.