Question #a505f

1 Answer
May 15, 2017

Answer:

#n^2# orbitals.

Explanation:

The principal quantum number, #n#, tells you the energy level on which an electron resides inside an atom.

The number of orbital present on a given energy level can be determined by looking at the value of the angular momentum quantum number, #l#, which tells you the subshell in which the electron resides.

figures.boundless.com

As you can see, for any value of #n#, you have

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

The number of orbitals present in a given subshell is given by the magnetic quantum number, #m_l#, which can take the values

#m_l = {-l, - (l-1), ..., -1, 0, +1, ..., +(l-1), + l}#

Now, for the first energy level, you have

#n=1#

This implies that

#l = 0 -># one subshell

and

#m_l = 0 -># one orbital

You can thus say that the first energy level holds #1# subshell and #1# orbital.

For the second energy level, you have

#n = color(red)(2)#

This implies

#l = {0, 1} -># two subshells

and

  • #l = 0 implies m_l = 0 -># one orbital
  • #l=1 implies m_l = {-1, 0 ,+1} -># three orbitals

Therefore, you can say that the second energy level holds #2# subshells and a total of #4# orbitals, so

#"no. of orbitals" = 4 = color(red)(2)^2#

For the third energy level, you have

#n = color(red)(3)#

This implies

#l = {0, 1, 2} -># three subshells

and

  • #l=0 implies m_l = 0 -># one orbital
  • #l=0 implies m_l = {-1, 0 ,+1} -># three orbitals
  • #l=2 implies m_l = {-2, -1, 0, +1, +2} -># five orbitals

You can thus say that the third energy level holds a total of #9# orbitals, so

#"mo. of orbitals" = 9 = color(red)(3)^2#

You can go on if you want, but at this point, it should be clear that the number of orbitals present for a given energy level, i.e. for a given value of the principal quantum number, is given by

#color(darkgreen)(ul(color(black)("no. of orbitals" = n^2)))#