Question

What characteristic is given by the angular momentum quantum number?

Answer 1

The angular momentum quantum number `l` describes the shape of an orbital. It corresponds to each orbital type, i.e. `(0,1,2,3, . . . , n-1) = (s, p, d, f, g, h, . . . )`.

Note that:

• `l_max = n - 1`
• `l = 0, 1, 2, 3, . . . , n - 1`
• `2l + 1` is the number of orbitals in a given `l` subshell.
• `l` is the number of angular nodes in an orbital.

So two examples:

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`1s` orbital:

• `bb(n = 1)` for the principal quantum number (read it off the orbital name).

Thus, `l_max = bb(l = 0)`. This also means there is no angular node in the `1s` orbital.

• `2l + 1 = 2(0) + 1 = bb(1)`. Thus, there is only one `1s` orbital.

That is also determined by realizing that for the magnetic quantum number, `m_l = {0}` due to `l = 0`. i.e. the only `1s` orbital there is has `l = 0`.

`2s, 2p` orbital(s):

• `bb(n = 2)` for the principal quantum number (read it off the orbital name).

Thus, `bb(l = 0, 1)`, and `l_max = 1`. This also means there is no angular node in the `2s` orbital, but `bb(1)` angular node in each `2p` orbital.

• For the `2s` orbital(s), `2l + 1 = 2(0) + 1 = bb(1)`. Thus, there is only one `2s` orbital.

That is also determined by realizing that for the magnetic quantum number, `m_l = {0}` due to `l = 0`. i.e. the only `1s` orbital there is has `l = 0`.

• For the `2p` orbital(s), `2l + 1 = 2(1) + 1 = bb(3)`. Thus, there are three `2p` orbitals.

That is also determined by realizing that for the magnetic quantum number, `m_l = {-1, 0 +1}` due to `l = 1`. i.e. each `2p` orbital uniquely corresponds to either `l = -1`, `0`, or `+1`. As there are three `m_l` values, there are three `2p` orbitals.