How many electrons are in #n = 2#? What about #n = 4, l = 3#? What about #n = 6, l = 2, m_l = -1#?

1 Answer
Nov 17, 2015

Answer:

Here's what I got.

Explanation:

Since the question is a bit ambiguous, I will assume that you're dealing with three distinct sets of quantum numbers.

In addition to this, I will also assume that you're fairly familiar with quantum numbers, so I won't go into too much details about what each represents.

figures.boundless.com

  • #1^"st"# set # -> n=2#

The principal quantum number, #n#, tells you the energy level on which an electron resides. In order to be able to determine how many electrons can share this value of #n#, you need to determine exactly how many orbitals you have in this energy level.

The number of orbitals you get per energy level can be found using the equation

#color(blue)("no. of orbitals" = n^2)#

Since each orbital can hold amaximum of two electrons, it follows that as many as

#color(blue)("no. of electrons" = 2n^2)#

In this case, the second energy level holds a total of

#"no. of orbitals" = n^2 = 2^2 = 4#

orbitals. Therefore, a maximum of

#"no. of electrons" = 2 * 4 = 8#

electrons can share the quantum number #n=2#.

  • #2^"nd"# set #-> n=4, l=3#

This time, you are given both the energy level, #n=4#, and the subshell, #l=3#, on which the electrons reside.

Now, the subshell is given by the angular momentum quantum number, #l#, which can take values ranging from #0# to #n-1#.

  • #l=0 -># the s-subshell
  • #l=1 -># the p-subshell
  • #l=2 -># the d-subshell
  • #l=3 -># the f-subshell

Now, the number of orbitals you get per subshell is given by the magnetic quantum number, #m_l#, which in this case can be

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

#m_l = {-3; -2; -1; 0; 1; 2; 3}#

So, the f-subshell can hold total of seven orbitals, which means that you have a maximum of

#"no. of electrons" = 2 * 7 = 14#

electrons that can share these two quantum numbers, #n=4# and #l=3#.

  • #3^"rd"# set #-> n=6, l=2, m_l = -1#

This time, you are given the energy level, #n=6#, the subshell, #l=2#, and the exact orbital, #m_l = 1#, in which the electrons reside.

Since you know the exact orbital, it follows that only two electrons can share these three quantum numbers, one having spin-up, #m_s = +1/2#, and the other having spin-down, #m_s = -1/2#.