Question

How many orbitals make up the 4d subshell?

Answer 1

The same number that makes up any individual `d` subshell... `5`. What are these five `m_l` values, specifically, for `d` orbitals?

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`d` orbitals have an angular momentum quantum number `l` of `2`:

`color(white)(.)ul(l" "" ""shape")`
`color(white)(.)0" "" "s`
`color(white)(.)1" "" "p`
`color(white)(.)2" "" "d`
`color(white)(.)3" "" "f`
`color(white)(.)4" "" "g`
`vdots" "" "vdots`

`l` has a range of `0, 1, 2, 3, . . . , n-1`. Its projection in the `z` direction is `m_l`, the magnetic quantum number, and `|m_l| <= l`, i.e.

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

You can see that there is an odd number of `m_l` values because this tells you that `m_l = 0, pm1, pm2, . . . , pml`. Hence, there are `2l+1` values of `m_l` for a GIVEN `l`.

Since `l = 2`...

`2(2) + 1 = bb"five"` `d` orbitals exist in one `d` subshell of any `n`.

And these orbitals are each given one value of `m_l` in the set of `m_l = {-2, -1, 0, +1, +2}`.