Question

• magnetic quantum number (m)
• magnetic spin number (mₛ)

atomic orbitals
sketch the shape of an s - orbital.

sketch the shape of the p - orbitals:

sketch the shape of the d - orbitals:

sketch the shape of the f - orbitals:

theoretically, higher orbitals do exist, g orbitals would have an angular momentum quantum number of l = 4, and there would be 9 orientations. unfortunately, we do not have enough elements on the periodic table to fill these orbitals.

Explanation:

1. s-orbital: It is a spherical region centered on the atomic nucleus, with uniform electron density in all directions.
2. p-orbitals: There are 3 dumbbell-shaped orbitals, oriented along the x, y, and z axes ($p_x$, $p_y$, $p_z$), each with two lobes on opposite sides of the nucleus.
3. d-orbitals: There are 5 orbitals: 4 have a cloverleaf shape ($d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$) with four lobes, and 1 ($d_{z^2}$) has a dumbbell shape with a ring (torus) around its midpoint along the z-axis.
4. f-orbitals: There are 7 complex, multi-lobed orbitals with more intricate, dispersed shapes, each having 8 lobes (or a combination of lobes and nodal structures) oriented in different 3D spatial directions.

Answer:

• s-orbital: A single, hollow sphere centered on the atomic nucleus.
• p-orbitals: Three separate dumbbell shapes, each aligned along one of the three Cartesian axes (x, y, z), with two equal lobes on either side of the nucleus.
• d-orbitals:

1. $d_{xy}$, $d_{xz}$, $d_{yz}$: Four-lobed clover shapes oriented between the Cartesian axes.
2. $d_{x^2-y^2}$: Four-lobed clover shape aligned along the x and y axes.
3. $d_{z^2}$: A dumbbell along the z-axis with a circular ring (torus) around the nucleus in the xy-plane.

• f-orbitals: Seven distinct, complex shapes with 8 lobes each (or combined lobe/nodal structures), spread across multiple spatial orientations with more complex nodal planes than lower orbitals.