THE MAIN QUANTUM NUMBERS OVERALL DEFINE AVAILABLE ORBITAL SUBSHELLS
#n# is the principal quantum number, indicating the quantized energy level, and corresponds to the period/row on the periodic table. #n = 1, 2, 3, . . . , N# where #N# is some large arbitrary integer in the real numbers.
#l# is the angular momentum quantum number, which corresponds to the shape of the orbital (#s#, #p#, #d#, #f#, #g#, #h#, etc). It is equal to #0, 1, . . . , n-1#.
It defines a unique subshell.
With #n = 2#, we have access to the #2s# and #2p# orbitals, since we assign #l = 0# for the #s# orbitals and #l = 1# for the #p# orbitals. (Similarly, we assign #l = 2# for the #d# orbitals, #l = 3# for the #f# orbitals, and so on alphabetically: #g#, #h#, etc.)
Note that with #n = 2# we do not get #l = 2# because #n - 1 = 2 - 1 = 1# for a max #l# of #1#.
THE MAGNETIC QUANTUM NUMBERS FURTHER DEFINE A SET OF (ORTHOGONAL) ORBITALS
With #l = 0,1#, we introduce the magnetic quantum number #m_l#, which takes on the values #0, pm1, . . . , pml#.
Therefore:
- #\mathbf(m_l = 0)# for #l = 0#. For each #m_l# value, we have one orbital that corresponds.
- #\mathbf(m_l = -1, 0, +1)# for #l = 1#. For each #m_l# value, we have one orbital that corresponds.
Furthermore, we have the spin quantum number #m_s# for each unique electron, which can be #pm"1/2"# and cannot be the same for two electrons with the same #m_l#.
That is, since all electrons in the same orbital have the same #n#, #l#, and #m_l#, they can only have #m_s = "+1/2"# show up once and #m_s = "-1/2"# show up once.
That means two electrons per orbital only.
TOTAL NUMBER OF ELECTRONS POSSIBLE
Finally, with the #s# and #p# subshells combined, we have:
#(1 + 3) cancel("orbitals") xx "2 electrons"/cancel("orbital")#
#= color(blue)(8)# #color(blue)("electrons allowed")#
which is what we expect for the octet rule for #n = 2#.
