Pauli Exclusion Principle

Introduction

The Pauli exclusion principle governs how identical electrons occupy quantum states. In chemistry's four-number description, no two electrons in one atom can have the same complete set n, l, mₗ, and mₛ. This single restriction produces the two-electron limit for each spatial orbital.

Core explanation

An orbital is fixed by the first three quantum numbers n, l, and mₗ. Two electrons sharing it therefore have identical spatial labels. To remain distinct, one must have mₛ = +1/2 and the other mₛ = −1/2. A third electron would have to duplicate one of those complete sets because no third electron spin projection exists. This is why one orbital holds at most two electrons with opposite spins.

In a box diagram, an orbital may be empty, contain one arrow, or contain an up/down pair. Two arrows in the same direction in the same box violate Pauli. A diagram with three arrows in one box also violates it. Two same-direction arrows in different boxes do not violate Pauli because their mₗ values differ; in fact, Hund's rule often favors that distribution among equal-energy orbitals.

Pauli combines with orbital count to give subshell capacities. One s orbital permits two electrons; three p orbitals permit six; five d orbitals permit ten; seven f orbitals permit fourteen. It also explains why atoms with many electrons must occupy progressively different states rather than all collapsing into one lowest-energy 1s state.

The principle is not merely a repulsive force preventing electrons from being physically close. Electrons can have overlapping spatial distributions while occupying different allowed states. Pauli is a rule about identical fermion quantum states. Electrostatic electron-electron repulsion is a separate interaction that also affects atomic energies.

For ions, the same principle applies. Removing an electron does not change the two-electron orbital capacity, and adding an electron requires an available distinct quantum state. Configuration questions therefore use Pauli whether the species is neutral or charged.

Step-by-step reasoning

1. Identify whether two electrons share one n,l,mₗ orbital. 2. If they do, compare their spin labels. 3. Permit a pair only for opposite mₛ values. 4. Reject any diagram duplicating all four labels or placing over two electrons in one orbital.

Visual explanation

Draw one orbital box containing ↑↓ as allowed. Cross out ↑↑ in the same box, but allow ↑ in one p box and ↑ in another p box.

Real-world analogy

Two records can share a folder and filename only if they have distinct version tags; here the two spin labels distinguish electrons with the same spatial orbital labels. The analogy is bookkeeping, not a literal mechanism.

Real-world example

Neutral helium has two electrons in 1s. Its orbital diagram is 1s [↑↓], with opposite spins permitting the shared spatial orbital without duplicating the complete quantum-number set.

Why?

Why can a p subshell hold six electrons? Its three spatial orbitals each allow two distinct spin states under Pauli, giving 3 × 2 = 6 electrons in total.

Common misconception

“Pauli requires all electrons in an atom to have opposite spins.” It forbids identical complete four-number sets; many electrons in different orbitals can share one spin projection.

Worked example

Consider two electrons labeled (2,1,0,+1/2) and (2,1,0,−1/2). They share one 2p orbital but differ in spin, so Pauli permits the pair. Changing the second spin to +1/2 would make all four values identical and would be forbidden in the same atom.

Quick check

1. Are ↑↑ arrows allowed in one orbital box? Answer: No. They would duplicate the spin state while sharing all spatial orbital labels.

Exam focus

State the four-number uniqueness rule, then derive two-per-orbital. Keep Pauli distinct from Hund's rule and from ordinary Coulomb repulsion.

Advanced insight

The Pauli rule reflects the antisymmetry of a many-electron wavefunction under exchange of two identical electrons. The four-number version is its practical orbital-model consequence for atomic configurations.

Summary

No two electrons in one atom may share all four quantum numbers. One spatial orbital therefore accommodates at most two electrons with opposite spin projections.

Practice questions

1. Can two electrons in different 2p orbitals both have mₛ = +1/2? Answer: Yes. Their mₗ values differ, so their four-number sets differ. 2. Why can 1s not hold three electrons? Answer: Only two spin values exist for its one spatial orbital, so a third would duplicate a state. 3. Is electron-electron electrostatic repulsion the Pauli principle? Answer: No. Repulsion is an interaction; Pauli is a restriction on identical-electron quantum states.