The Four Quantum Numbers — Complete Rules:

The principal quantum number n defines the electron shell. It takes positive integer values (1, 2, 3, ...). As n increases: orbital size increases (r ∝ $n^{2}$), energy increases toward zero (E ∝ −1/$n^{2}$), and the electron is more loosely bound. Shell capacity = 2$n^{2}$.

The azimuthal quantum number l defines the orbital type. For a given n, l can be 0, 1, 2, ..., (n−1). Critical rule: l is always strictly less than n. The mapping: l=0 (s, spherical, 1 orbital), l=1 (p, dumbbell, 3 orbitals), l=2 (d, cloverleaf, 5 orbitals), l=3 (f, complex, 7 orbitals). Subshell capacity = 2(2l+1).

The magnetic quantum number mₗ defines spatial orientation. For a given l: mₗ = −l, −(l−1), ..., 0, ..., +(l−1), +l. Total values = 2l+1 = number of orbitals in the subshell. The spin quantum number mₛ = +½ (spin-up) or −½ (spin-down).

Nodes Summary:

• Total nodes in any orbital = n − 1
• Angular (planar) nodes = l (depends only on subshell type)
• Radial (spherical) nodes = n − l − 1
• Examples: 1s: 0 total, 2s: 1 radial, 2p: 1 angular, 3s: 2 radial, 3p: 1+1, 3d: 2 angular

Electronic Configuration Rules: Aufbau (n+l rule): 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p Pauli: Each orbital holds maximum 2 electrons with opposite spins. Hund's: In degenerate orbitals (equal n+l), electrons occupy singly with parallel spins first.

Anomalous Configurations (must memorize):

• Cr (Z=24): [Ar] 3$d^{5}$ 4$s^{1}$ (NOT 3$d^{4}$ 4$s^{2}$) — half-filled d stabilization
• Cu (Z=29): [Ar] 3$d^{10}$ 4$s^{1}$ (NOT 3$d^{9}$ 4$s^{2}$) — fully-filled d stabilization

Cation Formation (NEET trap): When transition metals form cations, 4s electrons are removed FIRST:

• Fe: [Ar]3$d^{6}$4$s^{2}$ → $Fe^{2+}$: [Ar]3$d^{6}$ (removes 4$s^{2}$)
• Fe: [Ar]3$d^{6}$4$s^{2}$ → $Fe^{3+}$: [Ar]3$d^{5}$ (removes 4$s^{2}$ then one 3d)
• Mn: [Ar]3$d^{5}$4$s^{2}$ → $Mn^{2+}$: [Ar]3$d^{5}$

Unpaired Electrons (important for magnetism): N=3, O=2, Fe=4 unpaired, $Fe^{3+}$=5 unpaired (maximum for 3d ions), Cu=1 unpaired, Zn=0 (diamagnetic)