Question #65437

QUANTUM NUMBERS

The three main quantum numbers describe the shape, energy level, and orientation of the atomic orbitals.

There is a fourth (#m_s#) which describes the spin of an electron in an orbital, and it is purely based on the properties of the electron.

• #n# is the principal quantum number which describes the energy level.

#n = 1, 2, 3, . . . # and is always a positive integer.

• #l# is the orbital angular momentum quantum number which describes the shape of the orbital. #l = 0, 1, 2, 3, . . . , n-1#.

#l = 0, 1, 2, 3, ...# for #s, p, d, f, ...# orbitals, respectively.

• #m# (more specifically, #m_l#) is the magnetic quantum number, which corresponds to each unique orbital orientation. #m_l# takes on the set of integers from #-l# to #l#, i.e.

#m_l = {0, pm1, pm2, . . . , pml}#.

• #m_s# (see why we need to say #m_l#?) is the electron spin quantum number, which describes the spin of the electron. It is only #pm1/2#, independent of the other quantum numbers.

2P ATOMIC ORBITAL

For example, a #2p# orbital is said to have:

• #n = 2# since it's #color(highlight)(2)p#
• #l = 1# since it's #2color(highlight)(p)#
• #m_l = {0, pm1, . . . , pm l} = {-1, 0, +1}#
  (if #l = 2#, then #m_l = {-2, -1, 0, +1, +2}#)

If the orbital has an electron, the electron's #m_s# is either #+"1/2"# or #-"1/2"# for spin up and spin down, respectively. Technically, we just say #pm"1/2"# because it could take either configuration (neither is preferential over the other).

We cannot say which electron is which, because they are indistinguishable.

The #2p_z# orbital looks like this:
![)

• While #m_l = 0#, that corresponds to the #2p_z# orbital.
• While #m_l = -1#, that corresponds to the #2p_x# or #2p_y# orbital.
• While #m_l = +1#, that corresponds to the #2p_y# or #2p_x# orbital. (The uniqueness that #m_l = pm1# matters, but the choice of assigning either to #p_x# or #p_y# is arbitrary.)

Lastly, the observations you learn from General Chemistry describe the idea that electrons fill the orbitals:

• with the lowest energy first, in part due to #n# (Aufbau Principle)
• one at a time
• in such a way that #m_s# is not the same for any two electrons both in a doubly-occupied orbital (Pauli Exclusion Principle).