What is the maximum number of electrons that can be in each #d# subshell?

The #3d# orbitals are given by the first three quantum numbers:

• #n = 3#, the principal quantum number
• #l = 2#, the angular momentum quantum number
• #m_l = {-2,-1,0,+1,+2}#, the magnetic quantum number

Recall that a set of orbitals corresponding to a set of #m_l# values has a degeneracy of #2l+1#. That indicates the number of orbitals with shape #d# that are the same energy in an atom.

Since #d# orbitals have #l = 2#, there are

#2(2) + 1 = bb(five)#

#d# orbitals for a given quantum level #n#. As such, there are five #3d#, five #4d#, . . . etc. orbitals.

Electrons require a fourth quantum number, the spin quantum number #m_s# to fully describe their quantum state, via their spin of #pm1/2#.

Since

• no two electrons can share the same quantum state, i.e. no two electrons can have the same #n#, #l#, #m_l#, and #m_s#
• each orbital is completely specified by #n#, #l#, and #m_l#
• #m_s# can only take on two values

there can only be two electrons in each orbital (cf. Pauli Exclusion Principle). Therefore, the maximum number of electrons in five total #d# orbitals for a given #n# is

#"2 electrons" xx "5 (n-1)d orbitals" = bb(10)#