Question

How would you find how many orbitals a sublevel has?

Answer 1

You examine the range of allowed values of `m_l` for a given `l`.

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For an atomic orbital, the angular momentum quantum number `l` corresponds to the shape of the orbital, and we have:

`ul(l" "" ""shape/subshell")`
`0" "" "color(white)(.)s`
`1" "" "color(white)(.)p`
`2" "" "color(white)(.)d`
`3" "" "color(white)(.)f`
`4" "" "color(white)(.)g`
`5" "" "color(white)(.)h`
`6" "" "color(white)(.)i`
`7" "" "color(white)(.)k`
`vdots" "" "vdots`

`m_l`, the magnetic quantum number, is the z-projection of `l` in integer steps:

`m_l = {-l, -l+1, . . . , 0, . . . , l-1, l}`

Each `m_l` value corresponds to an orbital in the subshell defined by `l`. There are `2l+1` such values of `m_l` for a given `l`, and thus, there are `bb(2l+1)` orbitals in each subshell.

`ul(l" "" ""shape/subshell"" "" ""Number of Orbitals")`
`0" "" "color(white)(.)s" "" "" "" "" "" "" "2(0)+1=1`
`1" "" "color(white)(.)p" "" "" "" "" "" "" "2(1)+1=3`
`2" "" "color(white)(.)d" "" "" "" "" "" "" "2(2)+1=5`
`3" "" "color(white)(.)f" "" "" "" "" "" "" "2(3)+1=7`
`4" "" "color(white)(.)g" "" "" "" "" "" "" "2(4)+1=9`
`5" "" "color(white)(.)h" "" "" "" "" "" "color(white)(..)2(5)+1=11`
`6" "" "color(white)(.)i" "" "" "" "" "" "" "2(6)+1=13`
`7" "" "color(white)(.)k" "" "" "" "" "" "color(white)(..)2(7)+1=15`
`vdots" "" "vdots" "" "" "" "" "" "color(white)(.)vdots`