UPSC Civil Services (Mains), Chemistry optional · 2018 · Paper I

Q2(a)(ii). Radial, angular and total nodes

Question

Determine the number of radial nodes, angular nodes and total number of nodes in the following orbitals: 2p, 3s, 3p, 4d, 5f.

Radial nodes =n-l-1, angular nodes =l, total =n-1. 2p: 0, 1, 1; 3s: 2, 0, 2; 3p: 1, 1, 2; 4d: 1, 2, 3; 5f: 1, 3, 4.

A radial node is a sphere on which the radial function Rₙₗ(r)=0; there are n-l-1 of them. An angular node is a plane or cone on which the angular function Yₗₘ(θ,φ)=0; there are l of them.
The total number of nodes, n-1, rises with n; more nodes mean more curvature of ψ and so higher kinetic energy. Each orbital of a given n has the same total, shared differently between radial and angular nodes.
3s has two zeros between the peaks, 3p and 4d have one each.

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

A radial node is a sphere on which the radial function Rₙₗ(r)=0; there are n-l-1 of them. An angular node is a plane or cone on which the angular function Yₗₘ(θ,φ)=0; there are l of them.

Orbital	n, l	Radial n-l-1	Angular l	Total n-1	The angular nodes
2p	2, 1	0	1	1	one nodal plane (the xy plane for 2p_z)
3s	3, 0	2	0	2	none; two spherical nodes
3p	3, 1	1	1	2	one plane
4d	4, 2	1	2	3	two planes (two cones for d_(z²))
5f	5, 3	1	3	4	three planes or cones

The total number of nodes, n-1, rises with n; more nodes mean more curvature of ψ and so higher kinetic energy. Each orbital of a given n has the same total, shared differently between radial and angular nodes.
The figure shows the radial distribution: 3s has two zeros between the peaks, 3p and 4d have one each.

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