30.9 The Pauli Exclusion Principle

Atomic Structure and the Pauli Exclusion Principle

The Pauli exclusion principle is the rule that determines how electrons arrange themselves inside atoms. It explains why electrons don't all collapse into the lowest energy level, and it's the reason the periodic table has the structure it does.

Structure of Atoms

Atoms consist of three subatomic particles:

• Protons carry a positive charge and sit in the dense central nucleus. The number of protons defines the element (hydrogen has 1, helium has 2).
• Neutrons are electrically neutral and also reside in the nucleus. They contribute to the atom's mass but don't affect its chemical identity. Carbon-12 has 6 neutrons, while carbon-14 has 8.
• Electrons carry a negative charge and occupy regions around the nucleus called orbitals. In a neutral atom, the number of electrons equals the number of protons.

The Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. Since quantum numbers describe an electron's exact state, this means every electron in an atom must be in a unique state.

Four quantum numbers define each electron:

• Principal quantum number ($n$): Identifies the electron shell. $n = 1$ is closest to the nucleus, $n = 2$ is the next shell out, and so on. Higher $n$ means higher energy.
• Angular momentum quantum number ($l$): Identifies the subshell shape. $l$ ranges from 0 to $n - 1$. Values $l = 0, 1, 2, 3$ correspond to the s, p, d, f subshells.
• Magnetic quantum number ($m_l$): Identifies the orbital's orientation in space. $m_l$ ranges from $-l$ to $+l$. For a p subshell ($l = 1$), the possible values are $-1, 0, +1$, giving three distinct orbitals.
• Spin quantum number ($m_s$): Identifies the electron's intrinsic spin direction. Only two values are possible: $+\frac{1}{2}$ (spin-up) or $-\frac{1}{2}$ (spin-down).

Because spin has only two possible values, each orbital can hold at most two electrons, and those two must have opposite spins. That's the direct consequence of the exclusion principle.

The Pauli exclusion principle applies to all fermions (particles with half-integer spin, like electrons, protons, and neutrons). It does not apply to bosons (particles with integer spin, like photons).

Electron Shells and Subshells

Electron shells are labeled by the principal quantum number $n$:

• $n = 1, 2, 3, 4$ correspond to the K, L, M, N shells (K is closest to the nucleus)

Each shell contains subshells labeled by $l$:

• $l = 0$ → s subshell (spherical shape)
• $l = 1$ → p subshell (dumbbell shape)
• $l = 2$ → d subshell (cloverleaf shape)
• $l = 3$ → f subshell (complex shape)

The number of orbitals in a subshell is given by $2l + 1$, and since each orbital holds 2 electrons, the maximum number of electrons per subshell is $2(2l + 1)$:

The maximum number of electrons in an entire shell $n$ is $2n^2$. So the first shell holds 2, the second holds 8, the third holds 18, and so on.

Distribution in Atomic Orbitals

Electrons fill orbitals following three rules:

1. Aufbau principle: Electrons fill orbitals starting from the lowest energy and working upward. The filling order is: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow 4d \rightarrow 5p \rightarrow 6s \rightarrow 4f \rightarrow 5d \rightarrow 6p$ Notice that $4s$ fills before $3d$. This happens because $4s$ is slightly lower in energy in multi-electron atoms.

2. Pauli exclusion principle: No more than two electrons per orbital, and they must have opposite spins.

3. Hund's rule: When filling orbitals of equal energy (called degenerate orbitals, like the three p orbitals), electrons spread out one per orbital with parallel spins before any pairing occurs. This minimizes electron-electron repulsion.

Electron configuration notation uses the subshell label with a superscript for the number of electrons. For example, carbon has 6 electrons:

This tells you 2 electrons are in the 1s subshell, 2 in the 2s, and 2 in the 2p. By Hund's rule, those two 2p electrons occupy separate orbitals with parallel spins rather than pairing up in one orbital.

Periodic Table and Electron Configuration

The periodic table's structure directly reflects how electrons fill orbitals:

• Periods (rows) correspond to the highest principal quantum number $n$ being filled. Elements in period 3 have their outermost electrons in the $n = 3$ shell.
• Groups (columns) reflect the number of valence electrons, the electrons in the outermost occupied shell. Group 1 elements all have 1 valence electron; Group 2 elements have 2.

Valence electrons are what determine an element's chemical behavior. Elements in the same group share similar chemical properties because they have the same valence electron configuration. For example:

• Group 1 (alkali metals) like sodium ($1s^2 \, 2s^2 \, 2p^6 \, 3s^1$) all have one valence electron and are highly reactive.
• Group 18 (noble gases) like neon ($1s^2 \, 2s^2 \, 2p^6$) have completely filled valence shells and are chemically inert.

Periodic trends such as atomic size and ionization energy also arise from the pattern of electron shell filling. Without the Pauli exclusion principle forcing electrons into higher shells, none of this structure would exist.

Quantum Mechanical Considerations

The Pauli exclusion principle has a deeper quantum mechanical basis. Fermions obey Fermi-Dirac statistics, and the total wavefunction describing multiple fermions must be antisymmetric under particle exchange. This means that if you swap two identical fermions, the wavefunction changes sign. If two fermions had identical quantum numbers, the wavefunction would equal zero, which corresponds to a state that can't exist.

The exchange interaction arises from this antisymmetric requirement. It's not a force in the traditional sense, but it affects how electrons in atoms and molecules arrange themselves and contributes to phenomena like ferromagnetism.

One dramatic consequence is degeneracy pressure. Even at absolute zero, the exclusion principle prevents all electrons from occupying the same lowest-energy state. This creates an outward pressure that resists compression. In white dwarf stars, electron degeneracy pressure is what prevents gravitational collapse, supporting a stellar remnant roughly the size of Earth but with the mass of the Sun.