Questions: Atomic Orbitals: Shapes and Nodal Structure

Question 1

How many radial nodes and angular nodes does the 3p orbital have? What is the total number of nodes?

Model answer: For the 3p orbital, n = 3 and l = 1. Radial nodes = n − l − 1 = 3 − 1 − 1 = 1. Angular nodes = l = 1. Total nodes = n − 1 = 2.

The formula total nodes = n − 1 is universal for hydrogen-like orbitals. The split between radial and angular depends on l: angular nodes = l, radial nodes = (n − l − 1). The 3p orbital thus has one spherical radial shell and one planar angular node, giving it a dumbbell shape with an inner spherical node cutting through each lobe.

Question 2

The 2s and 2p orbitals are degenerate in hydrogen (same energy). Despite having the same n, they look completely different. What accounts for the difference in shape?

Model answer: The difference is in l: 2s has l = 0 (no angular nodes, spherically symmetric), while 2p has l = 1 (one angular nodal plane, dumbbell shaped). The total node count is the same (n − 1 = 1), but the 2s puts that node at a radial shell, while the 2p puts it as an angular plane. Energy in hydrogen depends only on n, but shape depends on l.

This is why the 2s and 2p are degenerate in hydrogen but not in multi-electron atoms (where electron-electron repulsion breaks the n-degeneracy). The angular node in 2p means 2p electrons have zero probability in the nodal plane, while 2s electrons are found at all angles but have a spherical gap at one radius.