Madelung Rule

The Madelung rule (also known as the n+l rule or the diagonal rule) is an empirical rule that describes the order in which atomic orbitals are filled with electrons in the ground state configuration of atoms.


Key Principle

  1. Electrons fill orbitals in order of increasing n+l, where n is the principal quantum number and l is the azimuthal quantum number
  2. When two orbitals have the same n+l value, the orbital with the lower n is filled first
  3. Named after Erwin Madelung, though the rule was actually formulated by others
  4. Provides a systematic way to determine electron configuration without solving the Schrödinger equation

Orbital Filling Order

  1. 1s (n+l = 1+0 = 1)
  2. 2s (n+l = 2+0 = 2)
  3. 2p (n+l = 2+1 = 3)
  4. 3s (n+l = 3+0 = 3) - filled after 2p due to lower n
  5. 3p (n+l = 3+1 = 4)
  6. 4s (n+l = 4+0 = 4) - filled before 3d due to lower n
  7. 3d (n+l = 3+2 = 5)
  8. 4p (n+l = 4+1 = 5) - filled after 3d due to lower n
  9. 5s (n+l = 5+0 = 5) - filled after 4p due to lower n
  10. 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p (continuing pattern)

Physical Basis

  1. The rule reflects the balance between orbital energy and electron-electron interactions
  2. Lower n+l values generally correspond to lower energy orbitals
  3. Explains why 4s fills before 3d despite 3d having a lower principal quantum number
  4. Accounts for the structure of the periodic table and transition metal configurations

Exceptions

  1. Some transition metals have exceptions (e.g., Cr, Cu) due to half-filled or fully-filled subshell stability
  2. Lanthanides and actinides show deviations due to f-orbital effects
  3. Heavier elements may not strictly follow the rule due to relativistic effects