Pauli exclusion principle

Question

if one subshell of an atom has 9 electrons in it, what is the minimum value of l? What is the spectroscopic notation for this atom, if this subshell is part of the n=3 shell?

Answer 1

the subshells are labeled

s,p,d,f,...

and numbered

l = 0,1,2,3,....respectively

each subshell contains

2(2l+1)

electrons, so they contain

2,6,10,14,......respectively

so to contain 9 electrons it must be at least the d subshell, ie l = 2

there are 4 pairs and one lone electron so it will have spin down (lower energy level than spin up) so

s = -½ is the spin number

then the total angular momentum quantum number is

j = l + s = 3/2

so the spectroscopic notation for the ground state is

3D(3/2)

where the 3/2 is a subscript with no brackets

.,..,

Answer 2

For any given l level you can have 2l+1 m sub l levels.  Each m sub l level can have two electrons by the exclusion principle.  Given 9 electrons, you can have 4 pairs with 1 left over.  So you need 5 m sub l levels.  Then:

5 = 2l + 1 --> l = (5-1)/2 = 4/2 = 2 so l = 2 is the smallest l value.

If n = 3, l =2, you have a 3P orbital