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

Q2(a)(i). Orbital and probability density

5 marks
Answered

Question

What is an orbital and how is it related to the probability density?

Answer in brief

An orbital is a one-electron wavefunction ψ_(n,l,mₗ)(r,θ,φ). Its square |ψ|² is the probability density: |ψ|²dτ is the probability of finding the electron in the volume element dτ.

Key points

An orbital is a solution ψ of the Schrödinger equation for one electron in an atom (atomic orbital) or a molecule (molecular orbital). It is fixed by the quantum numbers n, l, mₗ and holds at most two electrons of opposite spin.
ψ itself is an amplitude; it can be positive or negative and has no direct physical meaning. Its sign matters for overlap and bonding.
|ψ|² (or ψ*ψ) is the probability density. The probability of finding the electron in a small volume dτ at a point is |ψ|²dτ, and ∫|ψ|²dτ=1 over all space (normalisation).
Where ψ=0 (a node) the probability density is zero. The 2s orbital has a radial node at r=2a₀ in hydrogen.

The full model answer

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

An orbital is a solution ψ of the Schrödinger equation for one electron in an atom (atomic orbital) or a molecule (molecular orbital). It is fixed by the quantum numbers n, l, mₗ and holds at most two electrons of opposite spin.
ψ itself is an amplitude; it can be positive or negative and has no direct physical meaning. Its sign matters for overlap and bonding.
Born interpretation: |ψ|² (or ψ*ψ) is the probability density. The probability of finding the electron in a small volume dτ at a point is |ψ|²dτ, and ∫|ψ|²dτ=1 over all space (normalisation).
Where ψ=0 (a node) the probability density is zero. The 2s orbital has a radial node at r=2a₀ in hydrogen.
The radial distribution function P(r)=4π r²ψ² (for s orbitals) gives the probability of finding the electron in a thin shell at radius r. The picture of an orbital is a boundary surface enclosing about 90 % of the probability: an orbital is a region of high probability, not a path (orbit).

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Model answer to Q2(a)(i), 2018 Paper I: solution page 8 — Solution page 8 of 55

Read the whole 2018 solution 2018 question paper

All parts of Question 2

2(a)(i) Orbital and probability density5 marks
2(a)(ii) Radial, angular and total nodes5 marks
2(a)(iii) If Planck's constant were zero5 marks
2(b)(i) Significance of the Kelvin equation5 marks
2(b)(ii) Effect of temperature on viscosity5 marks
2(c) Bond orders of H₂⁺, Ne₂, CN⁻, NO and HF; order of stability15 marks
2(d)(i) Derivation of Bragg's equation5 marks
2(d)(ii) Number of K⁺ and Cl⁻ ions in 1.00 g of KCl5 marks