Is it true that for a value of #n# (n-principal quantum number), the value of #m_l# (#m_l#-magnetic quantum number) is equal to #n^2#? If it is true, can you please explain this to me? Thanks.

1 Answer
Jul 6, 2015

Answer:

The actual value, no. The number of values, yes.

Explanation:

The relationship between the principal quantum number, #n#, and the magnetic quantum number, #m_l#, actually goes through the angular momentum quantum number, #l#.

https://www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/introduction-to-quantum-theory-7

The angular momentum quantum number can take any value that ranges from #0# to #n-1#

#l = 0, 1, 2, ... , (n-1)#

The magnetic quantum number can take any value that ranges from #-l# to #l#

#m_l = -l, ... , -1, 0, 1, ... , l#

If you take into account the possible values of #l#, the magnetic quantum number can thus have values that range from

#m_l = -(n-1), ..., -1, 0, 1, ..., (n-1)#

So, for example, if #n=2#, #m_l# can be equal to

#m_l = 0# #-># for #l=0#

and

#m_l = {-1, 0, 1}# #-># for #l=1#

As you can see, the individual values of #m_l# don't even come close to the value of #n^2#.

However, the total number of values #m_l# can take, if you take into account the values of #l#, is indeed equal to #n^2#.

The magnetic quantum number actually tells you how many orbitals a subshell has. In the above example, if #n=2#, you get

#n=2#, #l=0#, #m_l = 0# #-># one 2s-orbital;

#{:(n=2, l=1, m_l = -1->2p_x), (n=2, l=1, m_l = 0->2p_y), (n=2, l=1, m_l = 1->2p_z) :}}# #-># three 3p-orbitals.

This means that the second energy level has a total of two subshells, the 2s-subshell and the 2p-subshell.

The number of orbitals each subshell contains is given by #m_l#. The 2s-subshell contains one orbital, since you only have one possible value for #m_l#.

The 2p-subshell contains 3 orbitals, each denoted by a different value of #m_l#.

The total number of orbitals in the second energy level is 4, which is equal to #n^2#.

If you try this with #n=3#, you'll find that you get a total number of #9# orbitals

  • one 3s-orbital;
  • three 3p-orbitals;
  • five 3d-orbitals.