QUANTUM NUMBERS

Overview of Quantum Numbers

  • Quantum numbers are necessary to describe the behavior and location of electrons in an atom.
  • Schrodinger's model allows for the representation of electrons in three-dimensional space, requiring three coordinates, or quantum numbers, to describe orbitals.
  • A fourth quantum number is introduced, the Spin quantum number (ms), which characterizes the specific behavior of a single electron in these orbital configurations.

Types of Quantum Numbers

1. Principal Quantum Number (n)
  • The principal quantum number (n) describes the size and energy of the orbital.
    • Larger values of n correspond to larger orbitals.
    • For instance, orbitals with n=2n = 2 are larger than those with n=1n = 1.
  • As n increases:
    • The orbital becomes larger.
    • The electron is found farther from the nucleus.
    • Higher energy is associated with these larger orbits, meaning the electron is less tightly bound to the nucleus.
  • Conceptual Summary: The principal quantum number essentially describes the electron shell or energy level of an atom.
2. Angular Momentum Quantum Number (l)
  • The angular momentum quantum number (l) defines the shape of the orbital and correlates to the angular momentum of an electron within that orbital.
  • Possible values for l range from 0 to n - 1 for every value of n.
  • Often referred to as subshell numbers and associated with specific orbital shapes:
    • l = 0: s orbital
    • l = 1: p orbital
    • l = 2: d orbital
    • l = 3: f orbital
    • l = 4: g orbital
  • Summary of Values and Corresponding Shapes:
    • s (spherical shape)
    • p (dumbbell shape)
    • d (complex shape)
    • f (more complex)
3. Magnetic Quantum Number (m)
  • The magnetic quantum number (m) specifies the orientation of the orbital in space.
  • The values of m can range from -l to +l, including zero, indicating the different orientations available within a subshell:
    • For s: 1 orbital (m = 0)
    • For p: 3 orbitals (m = -1, 0, +1)
    • For d: 5 orbitals (m = -2, -1, 0, +1, +2)
    • For f: 7 orbitals (m = -3, -2, -1, 0, +1, +2, +3)

Complete Electron Specification

  • The complete specification of an electron in any orbital can be described using the four quantum numbers: (n, l, m, ms).
  • The set of four quantum numbers is regarded as the address of an electron in an atom.

ORBITAL SHAPES

Orbital Shapes based on Quantum Numbers

  • The shapes of orbitals are determined by both the principal quantum number and the angular momentum quantum number:
1. S Orbitals
  • S orbitals are spherical in shape and their size increases with increasing n values.
    • Example of specific orbitals: 1s, 2s.
2. P Orbitals
  • P orbitals consist of two lobes separated by a node at the nucleus, characterized as having a dumbbell shape.
    • Specific orientations include: Px, Py, Pz.
3. D Orbitals
  • D orbitals have more complex shapes with two distinguished formations:
    • Four orbitals have four lobes centered in the plane.
    • The fifth orbital features two lobes with a "belt" around it.
    • Examples include: dxy, dxz, dyz, d2-y2, dz2.

ELECTRON CONFIGURATIONS

Electron Configuration Notations

  • Three methods can be utilized to write electron configurations for atoms:
    1. Orbital Diagrams: Visual representation of electrons in orbitals.
    2. spdf Notation: Descriptive notation detailing the specific subshell and number of electrons.
    3. Noble Gas Notation: Utilizes the electron configuration of the nearest noble gas to limit notation complexity.
Aufbau Principle
  • According to the Aufbau Principle, orbitals are filled in increasing order of energy levels.
    • Electrons fill the lowest energy orbitals first.
    • Example: 1s orbitals are filled before 2s orbitals.
Examples of Electron Configuration

  1. Carbon (C)

    • Total Electrons: 6
    • Full Configuration: 1s22s22p21s^2 2s^2 2p^2
    • Orbital Representation:

    \begin{array}{c}
    1s\,\uparrow\downarrow \
    2s\,\uparrow\downarrow \
    2p\,\uparrow\uparrow\downarrow
    \end{array}

Pauli's Exclusion Principle

  • Pauli Exclusion Principle states that no two electrons within an atom can have the same set of four quantum numbers.
    • This inherently implies that each electron occupies its own unique state.

Hund’s Rule

  • According to Hund’s Rule, electrons will occupy unoccupied orbitals first (degenerate orbitals) before pairing up in occupied orbitals.
  • This principle applies within subshells where the electrons exhibit the same energy levels, making them degenerate.

Shorthand Notation for Electron Configurations

  • Condensed Configurations: Writing shorthand electron configurations using noble gases to represent inner-core electrons.
    • Example:
    • Noble Gas Ne, Z = 10:
      • Full: 1s22s22p61s^2 2s^2 2p^6;
      • Shorthand: [Ne].
    • Sodium (Na), Z = 11:
      • Full: 1s22s22p63s11s^2 2s^2 2p^6 3s^1;
      • Shorthand: [Ne] 3s^1.
    • Scandium (Sc), Z = 21:
      • Full: 1s22s22p63s23p64s23d11s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1;
      • Shorthand: [Ar] 4s^2 3d^1.