What values of m are permitted for an electron with #l = 2#?

1 Answer
Jan 2, 2017

Answer:

Possible #m_l# values are #-2,-1,0,1,2#. See below.

Explanation:

There are four quantum numbers: the principle quantum number, #n#, the angular momentum quantum number, #l#, the magnetic quantum number, #m_l#, and the electron spin quantum number, #m_s#. For this question we are concerned with #l# and #m_l#.

The angular momentum quantum number, #l#, describes the shape of the subshell and its orbitals, where #l=0,1,2,3...# corresponds to #s, p, d, # and #f# subshells (containing #s, p, d, f# orbitals), respectively. Each shell has up to #n-1# types of subshells/orbitals.

An angular momentum quantum number of #l=2# describes a #d# subshell.

The magnetic quantum number, #m_l#, describes the orientation of the orbitals (within the subshells) in space. The possible values for #m_l# of any type of orbital (#s,p,d,f...#) is given by any integer value from #-l# to #l#.

Therefore, given #l=2#, the possible #m_l# values are #-2,-1,0,1,2#. This tells us that the #d# subshell contains five #d# orbitals, each with a different orientation (#d_(yz)#, #d_(xy)#, #d_(xz)#, #d_(x^2-y^2)#, and #d_(z^2)#).