Does the angular momentum quantum number #l# designate the shape of the orbital?

Basically yes. The angular momentum quantum number #l# corresponds to the shape of the orbital sublevel, and the magnetic quantum number #m_l# basically "builds" the shape when the #p# electron is subjected to a magnetic field.

But for the most part you can say that #l# indicates the shape because different values of #l# correspond to different orbital shapes.

If we take the #2p_z# orbital as an example and subject it to a magnetic field oriented in the #z# direction, the orbital angular momentum in the #z# direction sweeps out cones like this:

(Ignore the "#|uarr>>#" and "#|darr>>#"; it is not relevant, though it is called bra-ket notation if you look it up and bother to read more on it! Also, you would not have to explain this diagram on a General Chemistry test. In case you were curious though, the diagram shows #ℏ"/"2#, which is the height of the cones in units of #ℏ#. Anyways...)

Looks like a #2p_z# orbital, right? It should.

The orbital angular momentum in the #z# direction is #L_z#. In the following equation (which you don't need to use on a General Chemistry test):

#color(blue)(hatL_z)Y_l^(m_l)(theta,phi) = color(blue)(m_lℏ)Y_l^m(theta,phi)#

What you should notice here is that #hatL_z#, the orbital angular momentum operator for the z-direction, corresponds to #m_lℏ#.

If #n = 2#, #l = 1# corresponds to a #2p# atomic orbital, so #m_l = 0, pm1#.

• The #-1# refers to the lower cone on the above image.
• The #0# refers to a dot at the origin in the above image. It is also where the node is.
• The #+1# refers to the upper cone in the above image.

  This tells us that #m_l# is also known as the vector projection of #l#.

  Therefore, what you should notice is that #m_l# "builds" the shape of the orbital, while the number of #m_l# values corresponds to how many unique, orthogonal (perpendicular) orientations there are for any orbital in that sublevel.

  (The orthogonality is not crucial knowledge for General Chemistry, but it matters because a quantum mechanics postulate states that any orbital must be orthogonal to every other orbital in its sublevel.)

  Ultimately, since there are three values for #m_l#, there are three #2p# orbitals: #2p_x#, #2p_y#, and #2p_z#.