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Node questions look like free marks: two formulas, plug in n and l. Then the paper asks where a node is, or shows a graph and asks which orbital it belongs to, or asks how many nodal planes dz² has. This post covers the counting rules, finds node positions for hydrogen-like orbitals, explains the shapes of angular nodes and lists the mistakes that cost marks.
What are the rules, and which exams use them?
The NCERT Class 11 chapter Structure of Atom defines a node as a region where the probability density falls to zero. It gives:
Radial nodes: n − l − 1
Angular nodes: l
Total nodes: n − 1
The JEE (Main) 2026 syllabus lists the variation of ψ and ψ² with r for 1s and 2s orbitals, and the JEE (Advanced) 2026 brochure lists wave function and probability density, as plots only. Check the syllabus for your exam year.
How do you count nodes for any orbital?
Find l from the letter (s = 0, p = 1, d = 2, f = 3), then apply the rules.
1s: 0 radial, 0 angular
2s: 1 radial, 0 angular
2p: 0 radial, 1 angular
3p: 1 radial, 1 angular
3d: 0 radial, 2 angular
4d: 1 radial, 2 angular
4f: 0 radial, 3 angular
6d: 3 radial, 2 angular
A quick check: radial plus angular must equal n − 1. For 6d, 3 + 2 = 5 = 6 − 1.
Why do students find this harder than it looks?
Papaphotis and Tsaparlis (2008) tested 125 Grade 12 students on basic quantum chemistry. Performance was relatively high on questions needing recall or a routine procedure and much lower on questions needing conceptual understanding, and many answers drew on the Bohr model or mixed it with the quantum picture. Node counting is the routine part. The questions that separate students ask what a node is, so the rest of this post is about that.
Where exactly is a radial node?
A radial node is a sphere where the radial part of the wavefunction, R(r), is zero, not counting r = 0 or infinity. Using ρ = Zr/a₀, where a₀ is the Bohr radius (52.9 pm in NCERT), the hydrogen-like radial functions in this LibreTexts table contain these factors:
2s: (2 − ρ). Zero at ρ = 2.
3s: (27 − 18ρ + 2ρ²). Zero at two values of ρ.
Example 1: hydrogen 2s. Z = 1, so r = 2a₀ = 2 × 52.9 = 105.8 pm. A LibreTexts worked exercise reaches the same r = 2a₀.
Example 2: He⁺ 2s. Z = 2, so ρ = 2 means r = a₀ = 52.9 pm. A larger nuclear charge pulls the node inwards.
Example 3: hydrogen 3s. Solve 2ρ² − 18ρ + 27 = 0:
ρ = [18 ± √(324 − 216)] ÷ 4 = (18 ± 10.39) ÷ 4, so ρ = 1.90 or 7.10.
The nodes are at about 1.90 × 52.9 = 101 pm and 7.10 × 52.9 = 375 pm. Two radial nodes, as 3 − 0 − 1 = 2 predicts.
What do angular nodes look like?
Angular nodes come from the angular part of the wavefunction and pass through the nucleus.
pz: NCERT notes that the xy-plane is a nodal plane. One angular node.
dxy: two nodal planes, the xz- and yz-planes, as NCERT describes.
dx²−y²: two nodal planes, the planes x = y and x = −y, between the lobes.
dz²: no nodal planes at all. Its angular part contains (3cos²θ − 1), shown in the LibreTexts table of spherical harmonics. That is zero when cos²θ = 1/3, at θ ≈ 54.7° and 125.3° from the z-axis. Each is a cone, so dz² has two conical angular nodes.
So "angular nodes = l" is always true, but "angular nodes are planes" is not.
How do you read a radial graph in an exam?
On a graph of R against r, a radial node is where the curve crosses zero. On a graph of R² or the radial probability, it is where the curve touches zero between humps. Since there are n − l − 1 nodes, there are n − l humps. A graph with three humps and an s orbital means n − 0 = 3, so it is 3s. A graph that starts at zero at the nucleus is not an s orbital, because only s functions are non-zero at r = 0.
Common mistakes
Counting r = 0 or infinity as a node.
Using n − l instead of n − l − 1.
Saying dz² has two nodal planes.
Forgetting that a larger Z moves radial nodes closer to the nucleus.
Calling a node a place the electron "cannot cross". It is where the probability density is zero, not a wall.
For the quantum number rules these formulas rest on, see our post on quantum numbers and electron configuration. Nodes are easier to understand when you can turn a 3D shape around, and the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
A practice routine
Make the radial and angular count for every orbital from 1s to 5f and check each sum is n − 1.
Find the radial node of Li²⁺ 2s (Z = 3) in pm. You should get 2a₀/3, about 35.3 pm.
Sketch R² against r for 1s, 2s and 3s, marking humps and nodes.
Explain in one sentence why dz² still has l = 2 angular nodes.
Before your next test, write the three node rules at the top of your rough sheet and use them as a check.
