Quantum Mechanics 101 · Unit 4: Atoms, Spin & Identity

Quantum Academy · Chapter 18: Pauli Exclusion & the Periodic Table

Why atoms have shells, chemistry exists, and matter is solid

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Instructions. Work these with a pencil, a calculator and the n + l filling order (1s 2s 2p 3s 3p 4s 3d 4p). Notation: a configuration lists each occupied subshell with a superscript counting its electrons, so sodium is 1s² 2s² 2p⁶ 3s¹, abbreviated [Ne] 3s¹, where [Ne] stands for neon's closed-shell core. Take 13.6 eV as hydrogen's ionization energy, ħ = 1.055×10⁻³⁴ J·s, and 1 solar mass = 1.989×10³⁰ kg. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Pauli exclusion principle
2.Fermion
3.Electron configuration
4.Aufbau principle
5.Madelung rule
6.Hund's rule
7.Effective nuclear charge
8.Valence electrons
9.Ionization energy
10.Alkali metal
11.Noble gas
12.Degeneracy pressure

A.A group-1 element, lithium to francium, with a single s electron outside a closed-shell core; highly reactive.
B.Electrons occupy equal-energy orbitals singly with parallel spins before any orbital takes a pair.
C.Build an atom's ground state by adding electrons one at a time to the lowest-energy orbital still available.
D.The outermost electrons of an atom, which take part in chemical bonding and set its chemical behaviour.
E.No two identical fermions can occupy the same quantum state; in an atom, no two electrons share all four quantum numbers.
F.A particle with half-integer spin, such as an electron, proton, neutron or quark, which obeys the exclusion principle.
G.The list of occupied subshells and their electron counts, such as 1s² 2s² 2p⁶ 3s¹ (or [Ne] 3s¹) for sodium.
H.Pressure that arises because exclusion forces compressed fermions into high-momentum states; it persists even at absolute zero.
I.A group-18 element with completely filled outer s and p subshells (helium has only 1s²), and so very unreactive.
J.Subshells fill in order of increasing n + l; on a tie, the subshell with smaller n fills first.
K.The net positive charge Z_eff an electron feels after inner electrons shield part of the nucleus's charge Ze.
L.The minimum energy needed to remove an electron from an isolated atom or ion in its ground state; the first ionization energy removes the most loosely bound electron of the neutral atom.

Part B Questions

1.
Which statement is the Pauli exclusion principle for electrons in an atom?Circle one
1. ANo two electrons can have the same energy.
2. BNo two electrons can share all four quantum numbers n, l, mₗ and mₛ.
3. CNo two electrons can be in the same shell.
4. DElectrons with opposite spins repel each other more strongly.
2.
How many electrons can a complete d subshell hold?Circle one
1. A5
2. B6
3. C10
4. D18
3.
Which element is the first, in order of increasing Z, to have an electron in a 3d orbital in its ground state?Circle one
1. AArgon (Z = 18)
2. BPotassium (Z = 19)
3. CScandium (Z = 21)
4. DCalcium (Z = 20)
4.
Two electrons may occupy the same orbital provided their spins point the same way.Circle one
TrueFalse
5.
If a white dwarf cooled all the way to absolute zero, its electron degeneracy pressure would vanish and the star would collapse.Circle one
TrueFalse
6.
When electrons enter a set of equal-energy orbitals, they occupy them singly with parallel spins before pairing up. This is rule.Fill in the blank
7.
The outermost electrons, which take part in chemical bonding, are called electrons.Fill in the blank
8.
Using the n + l rule and the idea of penetration, explain why potassium's nineteenth electron enters 4s rather than 3d.Short answer
9.
Why does the fact that iron is much harder than solid neon show that 'full' atoms are not what makes matter solid? What is the real reason matter resists compression?Short answer
10.
Chromium's ground state is [Ar] 3d⁵ 4s¹. Write the configuration the simple n + l rule predicts, and give the number of unpaired electrons in each.Short answer
11.
The n = 4 shell can hold 2n² electrons. Calculate that number, then calculate how many elements the fourth row (period 4) of the periodic table actually contains, and explain the difference.Show your working
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12.
Potassium's first ionization energy is 4.34 eV, and its valence electron has n = 4. Use E ≈ 13.6 eV × Z_eff²/n² to estimate the effective nuclear charge it feels.Show your working
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Answer:
13.
A neutron star has a mass of 1.4 solar masses and a radius of 12 km. Calculate its mean density and compare it with the mean density of Sirius B, 2.7×10⁹ kg/m³.Show your working
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14.
For slow (non-relativistic) electrons, degeneracy pressure scales as nₑ to the 5/3 power. A white dwarf is compressed to half its radius at fixed mass. By what factor does its electron density rise, and by what factor does its degeneracy pressure rise? What would the pressure factor be if the electrons were ultra-relativistic, with pressure scaling as nₑ to the 4/3 power?Show your working
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Answer:

Part C Mad scientist challenge bonus

Prove that white dwarfs have a maximum mass, using nothing but scaling. (a) Hydrostatic balance needs a central pressure of order P ~ GM²/R⁴. Non-relativistic degeneracy pressure scales as nₑ to the 5/3 power, with nₑ ∝ M/R³. Show that balance requires R ∝ M to the −1/3 power. What happens to the star as you add mass? (b) For ultra-relativistic electrons the pressure scales as nₑ to the 4/3 power. Show that R cancels out of the balance, and explain why that implies a maximum mass. (c) Evaluate the natural mass scale M₀ = (ħc/G) to the 3/2 power, divided by (2mᵤ)², using ħc = 3.16×10⁻²⁶ J·m, G = 6.674×10⁻¹¹ N·m²/kg² and mᵤ = 1.661×10⁻²⁷ kg (the 2 is two nucleons per electron). The full stellar-structure calculation multiplies M₀ by about 3.10. What limiting mass results, in solar masses?

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