UPSC Civil Services (Mains), Chemistry optional · 2026 · Paper I

Q7(b)(ii). Quantum numbers of the 2p_z orbital

Question

What are the n, l and mₗ values of 2p_z orbital? Explain.

n=2, l=1, mₗ=0

The number 2 is the principal quantum number: n=2, the second shell (L shell). It fixes the energy (in hydrogen) and the size of the orbital.
For n=2, l=0 or 1. The letter p means l=1 (azimuthal quantum number). It fixes the shape and the orbital angular momentum, √(l(l+1)) ħ=√2 ħ.
three p orbitals. The function with mₗ=0 is ψ=R₂₁(r) Y_(1,0)∝ rθ f(r)=z f(r), which points along z. Hence 2p_z: n=2, l=1, mₗ=0
pₓ and p_y are not single mₗ states. They are the sum and the difference of the mₗ=+1 and -1 functions, so only p_z has a definite mₗ. The 2p_z electron has zero angular momentum about the z axis.

The full model answer

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

The number 2 is the principal quantum number: n=2, the second shell (L shell). It fixes the energy (in hydrogen) and the size of the orbital.
For n=2, l=0 or 1. The letter p means l=1 (azimuthal quantum number). It fixes the shape and the orbital angular momentum, √(l(l+1)) ħ=√2 ħ.
For l=1, mₗ=+1,0,-1: three p orbitals. The function with mₗ=0 is ψ=R₂₁(r) Y_(1,0)∝ rθ f(r)=z f(r), which points along z. Hence 2p_z: n=2, l=1, mₗ=0
pₓ and p_y are not single mₗ states. They are the sum and the difference of the mₗ=+1 and -1 functions, so only p_z has a definite mₗ. The 2p_z electron has zero angular momentum about the z axis.
Nodes: n-1=1 in all. Angular nodes =l=1: the xy plane, where z=0 and ψ changes sign. Radial nodes =n-l-1=0. The orbital is a dumbbell along z, with lobes of opposite sign.

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