What is the azimuthal quantum number (also called the angular-momentum quantum number), ℓ, for the orbital shown here?

(the photo is a d orbital. it has 4 balls around it)

Express your answer numerically as an integer.

l=2

I can't see the photo but a d orbital has a value of 2 for ell (I can't write the script ell and the lower case l looks like (OK, ALMOST like) the number 1).(see 1 vs l)

s orbitals have ell = 0
p orbitals have ell = 1
d orbitals have ell = 2

Ah, a delightful d orbital! Just like my clown nose, it's a spectacle to behold! Now, to answer your question, the azimuthal quantum number (ℓ) for a d orbital is 2. So, grab some popcorn and enjoy the show!

To determine the azimuthal quantum number (ℓ) for the given orbital, you can refer to the shape of the orbital itself. In this case, you mentioned that the orbital has four balls around it, which indicates it is a d orbital.

The d orbitals have various shapes depending on their sublevel, labeled as dxy, dxz, dyz, dx2-y2, and dz2. Each shape corresponds to a different value of ℓ.

By considering the description of the orbital as having 4 balls around it, it suggests that the orbital is likely the dz2 orbital. The dz2 orbital has a double-donut shape, with the lobes pointing along the z-axis and a ring in the xy-plane.

For the dz2 orbital, the value of ℓ is 2. Therefore, the azimuthal quantum number (also known as the angular-momentum quantum number) for the given orbital is ℓ = 2.