PHYC10004 Chap.12 Quantum Numbers, Spin and the Periodic Table
Quantum Numbers, Spin and the Periodic Table
Week 10 labels every electron state with four quantum numbers. The principal number n sets the shell, the orbital number l from 0 to n − 1 sets the orbital angular momentum L = √(l(l+1)) ħ, the magnetic number m_l from −l to +l sets its component along an axis, and the spin number m_s = ±½ completes the set.
Counting the allowed combinations gives n² orbitals and 2n² electron states in shell n.
The Pauli exclusion principle, that no two electrons share all four numbers, turns this counting into the structure of the periodic table: closed shells such as helium and neon are hard to ionise, while alkali metals hold one loosely bound outer electron.
Moseley's measurements of X-ray lines, whose frequencies scale with (Z − 1)², ordered the elements by atomic number. The chapter ends with Heisenberg's uncertainty principle in the form printed on the 2026 formula sheet, Δx Δp ≥ ħ/2, and its consequences for confined particles.
What this chapter covers
- 01
Principal Quantum Number
- 02
Orbital Angular Momentum
- 03
Magnetic Quantum Number
- 04
Electron Spin
- 05
Pauli Exclusion Principle
- 06
Shell Filling and the Periodic Table
- 07
Ionisation Energy Trends
- 08
Moseley's Law
- 09
Heisenberg Uncertainty Principle
Worked example · free
Count the states in the n = 4 shell
- 1Allowed l values run from 0 to n − 1: l = 0, 1, 2 and 3.
- 1Each l has 2l + 1 values of m_l: 1 + 3 + 5 + 7 = 16 orbitals, which is n².
- 1Two spin states per orbital give 2n² = 32 electron states.
- 1Largest L, for l = 3: L = √(3 × 4) ħ = 3.46ħ = 3.65 × 10⁻³⁴ J s.
Key terms
- Principal Quantum Number
- The principal quantum number n = 1, 2, 3 labels the shell of an electron state and, in hydrogen, fixes its energy.
- Orbital Quantum Number
- The orbital quantum number l, from 0 to n − 1, fixes the size of an electron's orbital angular momentum, L = √(l(l+1)) ħ.
- Magnetic Quantum Number
- The magnetic quantum number m_l, from −l to +l, fixes the component of orbital angular momentum along a chosen axis, L_z = m_l ħ.
- Electron Spin
- Electron spin is an intrinsic angular momentum whose component along an axis takes only the values m_s ħ with m_s = +½ or −½.
- Pauli Exclusion Principle
- The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers.
- Uncertainty Principle
- The uncertainty principle states that the uncertainties in position and momentum satisfy Δx Δp ≥ ħ/2, so both cannot be known exactly at once.
Quantum Numbers, Spin and the Periodic Table FAQ
How many electrons can each shell hold?
Shell n contains n² orbitals, because each value of l from 0 to n − 1 has 2l + 1 values of m_l. Each orbital holds two electrons with opposite spin, so the shell holds 2n² electrons: two, eight, eighteen and thirty-two for the first four shells.
Why is sodium's outer electron easy to remove?
Sodium's eleventh electron sits alone in the n = 3 shell, outside two closed shells. The ten inner electrons screen most of the nuclear charge, so the outer electron feels roughly the attraction of a single proton and needs only about five electronvolts to escape.
What did Moseley's X-ray measurements show?
The square root of the Kα line frequency rises linearly with atomic number, because the falling electron sees an effective nuclear charge of Z − 1. Ordering elements by these lines put the periodic table in order of atomic number rather than atomic mass.
Which form of the uncertainty principle should I use?
Use the form printed on the 2026 formula sheet, Δx Δp ≥ ħ/2. Past solutions have used h, h/2 or ħ/2 and noted that each gives a different number, so always state which form you are applying in your working.
Exam move
Make a table of the four quantum numbers with their allowed values, meanings and formula-sheet relations, and practise listing every allowed set for a given shell or sub-shell. Fill orbital boxes for the first eleven elements and explain the ionisation-energy jumps from helium to lithium and from neon to sodium.
Work Moseley calculations with the screening argument written in words, and finish with uncertainty-principle estimates for an electron in an atom, stating the form of the principle you use.
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