Quantum Mechanical Model of the Atom - Energies of Orbitals

Aufbau Principle:

Electrons fill orbitals in order of increasing energy. The energy order follows the (n + l) rule:

• n + l rule: Lower value of (n + l) → lower energy. If (n + l) is the same for two orbitals, the one with lower n has lower energy.

• Filling order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s …

Hund's Rule of Maximum Multiplicity:

Electrons never pair up in orbitals of the same subshell until each orbital in that subshell has at least one electron (is singly occupied).

Pauli's Exclusion Principle:

No two electrons in an atom can have the same set of all four quantum numbers. As a result, each orbital can hold a maximum of 2 electrons with opposite spins.

Special Stability: Cr and Cu

• Chromium (Cr, Z=24): Expected 1s²2s²2p⁶3s²3p⁶3d⁴4s² → Actual: 1s²2s²2p⁶3s²3p⁶3d⁵4s¹ (half-filled 3d is extra stable)
• Copper (Cu, Z=29): Expected 1s²2s²2p⁶3s²3p⁶3d⁹4s² → Actual: 1s²2s²2p⁶3s²3p⁶3d¹⁰4s¹ (fully-filled 3d is extra stable)

Half-filled and fully-filled sets of degenerate orbitals have extra stability.

Nodes in Orbitals:

• Radial nodes = n − l − 1
• Angular nodes = l
• Total nodes = n − 1
• Number of nodal planes = l