Quantum Numbers in Atomic Structure

Quantum Numbers in Atomic Structure

The relation of a particular electron to the nucleus can be described through a series of four numbers, called the Quantum Numbers. The first three describe the energy, shape, and orientation of the orbital. The fourth represents the "spin" of the electron.

1. Principal Quantum Number (n)

  • Symbol: n
  • Values: n = 1, 2, 3, \ldots
  • Significance:
    • Specifies the energy of an electron and the size of the orbital.
    • All orbitals with the same n are in the same shell.
    • Higher n → higher energy, larger orbital, electron less tightly bound.

2. Angular Momentum Quantum Number (l)

  • Symbol: l
  • Values: 0, 1, 2, \ldots, n-1
  • Significance:
    • Defines the shape of the orbital.
    • Divides shells into subshells.
    • Letter codes:
      • l = 0 → s orbital (spherical)
      • l = 1 → p orbital (dumbbell-shaped)
      • l = 2 → d orbital (complex shape)
      • l = 3 → f orbital
    • Energy order: s < p < d < f

3. Magnetic Quantum Number (ml)

  • Symbol: m_l
  • Values: -l, \ldots, 0, \ldots, +l
  • Significance:
    • Specifies the orientation of the orbital in space.
    • Each subshell has 2l + 1 orbitals.
    • Example: p subshell (l = 1) has 3 orbitals: m_l = -1, 0, +1.

4. Spin Quantum Number (ms)

  • Symbol: m_s
  • Values: +\frac{1}{2} (spin up), -\frac{1}{2} (spin down)
  • Significance:
    • Describes the intrinsic spin direction of the electron.
    • Each orbital can hold 2 electrons with opposite spins.

Electron Configuration

  • Describes the distribution of electrons in orbitals.
  • Valence electrons determine chemical properties.
  • The periodic table is divided into blocks:
    • s-block: Groups 1 & 2 (Alkali & Alkaline Earth metals)
    • p-block: Groups 13–18 (Main group elements)
    • d-block: Groups 3–12 (Transition metals)
    • f-block: Lanthanides & Actinides

Rules for Filling Orbitals

1. Aufbau Principle

Electrons fill orbitals in order of increasing energy:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

2. Pauli Exclusion Principle

No two electrons can have the same four quantum numbers → each orbital holds max 2 electrons with opposite spins.

3. Hund’s Rule

Electrons fill degenerate orbitals singly before pairing up.

Example: Nitrogen (Z=7)

Configuration: 1s^2 \; 2s^2 \; 2p^3

p orbitals are half-filled with one electron each before pairing.

Periodic Table & Orbital Blocks

  • s-block: ns^1 to ns^2
  • p-block: np^1 to np^6
  • d-block: (n-1)d^1 to (n-1)d^{10}
  • f-block: (n-2)f^1 to (n-2)f^{14}

Magnetic Behavior

  • Diamagnetic: All electrons paired → not attracted to magnets.
  • Paramagnetic: Unpaired electrons present → weakly attracted to magnets.

Summary

  • n → shell & energy
  • l → subshell & shape
  • ml → orbital orientation
  • ms → electron spin
  • Follow Aufbau, Pauli, Hund’s rules for electron arrangement.