Question #10924

1 Answer
Aug 16, 2017

Answer:

Here's what I got.

Explanation:

I'm guessing that you want to figure out a possible set of quantum numbers that can describe an electron located in the #5f# subshell.

For starters, you should know that we can use a total of #4# quantum numbers to describe the location and spin of an electron inside an atom.

figures.boundless.com

Now, the subshell in which an electron resides is given by the angular momentum quantum number, #l#, which can take on of the following values

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

and so on. So in your case, all the electrons that can reside in the #5f# subshell will have

#l = 3#

Notice that the value of the angular momentum quantum number depends on the value of the principal quantum number, #n#.

This means that in order for your electrons to have access to the #color(red)(5)f# subshell, they need to be located on the fifth energy level, so

#n = color(red)(5)#

The #5f# subshell contains a total of #7# orbitals, all described by a distinct value of the magnetic quantum number, #m_l#.

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

http://boomeria.org/chemtextbook/cch9.html

The spin quantum number, #m_s#, which describes the spin of the electron inside its orbital, can only take two possible values

#m_s = { -1/2, + 1/2}#

Since each orbital can hold a maximum of #2# electrons of opposite spins, i.e. of opposite #m_2# values--think Pauli's Exclusion Principle here--you can say that the #5f# subshell can hold a maximum of

#7 color(red)(cancel(color(black)("orbitals"))) * "2 e"^(-)/(1color(red)(cancel(color(black)("orbital")))) = "14 e"^(-)#

This implies that you can write a total of #14# distinct quantum number sets to describe one of the #14# electrons that can share the #5f# subshell.

For example, you can have

  • #n =5 , l= 3 , m_l = -3, m_s = +1/2#
  • #n = 5, l = 3, m_l = 0, m_s = -1/2#
  • #n = 5, l = 3, m_l = 2, m_s = +1/2#
  • #n = 5, l = 3, m_l = -1, m_s = -1/2#

and so on. Notice that all the #14# sets share the principal quantum number and the angular momentum quantum number, which is what you should expect for electrons that share a subshell--the #f# subshell-- located on a specific energy level--the fifth energy level.