Question 1

What is the difference between a radial node and an angular node?

Accepted Answer

A radial node is a spherical surface at a fixed distance from the nucleus where the probability of finding the electron is zero — it depends only on distance (r). An angular node is a flat plane (or, for d_z², a cone) passing through the nucleus where the probability is zero — it depends only on direction (angle). Radial nodes = n − ℓ − 1, angular nodes = ℓ.

Question 2

How many nodes does a 3s orbital have?

Accepted Answer

A 3s orbital has 2 radial nodes and 0 angular nodes, giving 2 total nodes (n − 1 = 3 − 1 = 2). Both are spherical shells nested inside the orbital, at roughly ρ = 1.9 a₀ and ρ = 7.1 a₀, where the electron probability drops to exactly zero.

Question 3

Why does the d_z² orbital have conical nodes instead of planar nodes?

Accepted Answer

The d_z² angular function is proportional to (3cos²θ − 1), which equals zero when cos²θ = 1/3, i.e. at θ = 54.74° and 125.26° from the z-axis. These fixed angles trace out two cones (an hourglass shape) meeting at the nucleus — not flat planes. It still has 2 angular nodes, matching ℓ = 2, just in a conical form.

Question 4

What is the total number of nodes in any orbital?

Accepted Answer

The total number of nodes in any orbital is n − 1, where n is the principal quantum number. This splits into radial nodes (n − ℓ − 1) and angular nodes (ℓ). For example, a 4f orbital has n − 1 = 3 total nodes: 3 angular (ℓ = 3) and 0 radial.

Orbital Nodes — Radial, Angular & Conical in 3D

Practice

Learned it? Now prove it under the clock.