Pauli’s Exclusion Principle – Detailed Notes
1. Background
- Before 1925, the Bohr model explained the hydrogen atom successfully, but it failed for multi-electron atoms.
- In 1926, Schrödinger’s wave equation introduced the Quantum Mechanical Model of the Atom, where electrons are described as wave functions (ψ).
- Solutions of the Schrödinger equation gave rise to the concept of orbitals – regions in space where the probability of finding an electron is maximum.
2. Pauli’s Exclusion Principle (1925)
- Statement: No two electrons in an atom can have the same set of four quantum numbers (n, ℓ, mℓ, ms).
- This means every electron is unique inside an atom.
- If one electron has a given set of values for n, ℓ, mℓ, and ms, no other electron in the atom can have the exact same four values.
3. Explanation with Orbitals
- Each orbital is defined by three quantum numbers (n, ℓ, mℓ).
- A maximum of two electrons can occupy an orbital.
- To obey Pauli’s principle, these two electrons must differ in spin quantum number (ms = +½ or –½).
- Example: In the 1s orbital of Helium (1s²), both electrons occupy the same orbital (n=1, ℓ=0, mℓ=0), but one has ms = +½ and the other ms = –½.
4. Relation with Quantum Numbers
There are four quantum numbers that describe each electron:
-
Principal Quantum Number (n):
- Indicates shell/energy level (n = 1, 2, 3, …).
- Larger n → electron farther from nucleus → higher energy.
-
Azimuthal Quantum Number (ℓ):
- Describes subshell and orbital shape.
- Values: 0 to (n – 1).
- ℓ = 0 (s), ℓ = 1 (p), ℓ = 2 (d), ℓ = 3 (f).
-
Magnetic Quantum Number (mℓ):
- Orientation of orbital in space.
- Values: –ℓ to +ℓ.
- Example: for p (ℓ=1), mℓ = –1, 0, +1 (3 orbitals).
-
Spin Quantum Number (ms):
- Describes electron spin.
- Values: +½ (clockwise spin) or –½ (anticlockwise spin).
⚡ According to Pauli’s principle → No two electrons can have the same combination of these four numbers.
5. Importance of Pauli’s Principle
- Explains the structure of the periodic table.
- Governs the electronic configuration of elements.
- Determines the capacity of orbitals, subshells, and shells:
- One orbital → 2 electrons
- s subshell → 1 orbital → 2 electrons
- p subshell → 3 orbitals → 6 electrons
- d subshell → 5 orbitals → 10 electrons
- f subshell → 7 orbitals → 14 electrons
6. Wave Function and Symmetry
- Electrons are identical particles (fermions, spin = ½).
- Their wave functions are antisymmetric:
- If we exchange two electrons, the wave function changes sign.
- This antisymmetry leads to Pauli’s principle → electrons cannot have the same state.
- In contrast, particles with integer spin (bosons, e.g., photons) have symmetric wave functions and can occupy the same state (this is why many photons can be in the same laser beam).
7. Generalized Pauli Exclusion Principle
- Applies to all fermions (electrons, protons, neutrons, quarks, leptons).
- Fermions obey Fermi–Dirac statistics and cannot occupy the same state.
- Bosons (particles with integer spin) obey Bose–Einstein statistics and can occupy the same state.
8. Example – Electronic Configurations
- Hydrogen (Z = 1): 1s¹ → only one electron.
- Helium (Z = 2): 1s² → two electrons in same orbital but opposite spins.
- Lithium (Z = 3): 1s² 2s¹ → third electron enters new shell (2s).
- This explains why elements differ in chemical properties.
9. Summary
- Pauli’s Principle: No two electrons in an atom have the same four quantum numbers.
- Each orbital holds a maximum of 2 electrons with opposite spins.
- Explains atomic structure, periodic table, and electronic configurations.
- Applies to all fermions (spin = ½).
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