What are the quantum numbers for atoms?
1 Answer
There is actually a lot of memorization involved here. I would recommend you try flash cards, and write down these relations:
- The total number of orbitals in a subshell is equal to
#2l + 1# . - The total number of orbitals in one quantum level is
#n^2# . - The maximum number of electrons per orbital is
#2# . - The total number of radial nodes (spherical nodal shells) is
#n - l - 1# . - The angular momentum (azimuthal) quantum number
#l# is the total number of angular nodes (nodal planes). Another way to say it is that the number of nodal planes is#l# .
Along with those overarching rules that arise from the definitions of the quantum numbers, you should know what the quantum numbers directly tell you.
- The principal quantum number
#n# tells you what quantum level you are on.
#n = 1, 2, 3, . . . # So, at
#n = 4# , you are looking at an element from the fourth row on the periodic table.
- The angular momentum (azimuthal) quantum number
#l# tells you what the shape of the orbital is.
#l = 0, 1, 2, . . . , n - 1# , and each#l# corresponds to an orbital shape.
#(0,1,2,3,4, . . . ) harr (s,p,d,f,g, . . . )# .Thus,
#l# CANNOT be equal to#n# . That is why the#1p# ,#2d# ,#3f# ,#4g# , [...] orbitals do not exist.
- The magnetic quantum number
#m_l# represents each actual orbital in the subshell. So,#m_l = {-l,-l+1, . . . , 0, . . . , l - 1, l}# .
Thus,
#|m_l| <= l# , and#m_l# cannot be greater in magnitude than#l# . For example, a#p# subshell, with#l = 1# , would have three orbitals. That is because their#m_l = {-1,0,+1}# , so there are three orbitals, corresponding to three total#m_l# values.
- The spin quantum number
#m_s# is simple; it is only#pm"1/2"# for electrons, no exceptions.
PRACTICE PROBLEM
What is the set of quantum numbers corresponding to a single, spin-up electron in a
ANSWER:
#color(white)(n = 4)# #color(white)(", because the")# #color(white)(4)# #color(white)("in front tells you what")# #color(white)(n)# #color(white)("is.")# #color(white)("l = 1")# #color(white)(", because")# #color(white)(l = 1)# #color(white)("corresponds to a")# #color(white)(p)# #color(white)("subshell.")# #color(white)(m_l = +1)# #color(white)(", because we've accounted for")# #color(white)(m_l = 0,-1)# #color(white)("already, and")# #color(white)(m_l = {-1,0,+1})# #color(white)("since")# #color(white)(l = 1)# #color(white)("tells you the range of"# #color(white)(m_l)# #color(white)("values allowed.")# #color(white)(m_s = +"1/2")# #color(white)(", because a spin-up electron has a")#
#color(white)("positive spin quantum number value.")#
(highlight to see)
