Quantum Numbers Lecture Notes

Overview

• Quantum numbers describe the address of an electron in an atom.
• Based on the Pauli exclusion principle, each electron has a unique set of four quantum numbers.

Principal Quantum Number (n)

• Symbol: n
• Integer values: 1, 2, 3, 4, etc.
• Describes the energy level of an electron in an atom
• Example:
  • 1st shell: n = 1
  • 2nd shell: n = 2
  • 3rd shell: n = 3

Angular Momentum Quantum Number (l)

• Symbol: l
• Describes the shape of the sublevel within an energy level
• Sublevels and corresponding l values:
  • s sublevel: l = 0 (sphere)
  • p sublevel: l = 1
  • d sublevel: l = 2
  • f sublevel: l = 3
• Relationship between n and l:
  • l ≤ n - 1
  • Example ranges for l:
    • n = 1: l = 0
    • n = 2: l = 0, 1
    • n = 3: l = 0, 1, 2
    • n = 4: l = 0, 1, 2, 3
• Notable configurations:
  • No 1p sublevel
  • No 2d sublevel
  • Possible 4f sublevel
• Example calculation for electron in sublevel:
  • 3d sublevel: n = 3, l = 2
  • 4f sublevel: n = 4, l = 3

Magnetic Quantum Number (m_l)

• Symbol: m_l
• Describes the orbital within a sublevel
• Relationship between l and m_l:
  • m_l ranges from -l to +l
  • Example range for m_l:
    • l = 0: m_l = 0
    • l = 1: m_l = -1, 0, 1
    • l = 2: m_l = -2, -1, 0, 1, 2
    • l = 3: m_l = -3, -2, -1, 0, 1, 2, 3
• Orbital capacities:
  • s sublevel (l = 0): 1 orbital, m_l = 0
  • p sublevel (l = 1): 3 orbitals, m_l = -1, 0, 1
  • d sublevel (l = 2): 5 orbitals, m_l = -2, -1, 0, 1, 2
  • f sublevel (l = 3): 7 orbitals, m_l = -3, -2, -1, 0, 1, 2, 3

Electron Spin Quantum Number (m_s)

• Symbol: m_s
• Describes the spin of an electron
• Possible values: +1/2, -1/2
• Upward spin: +1/2
• Downward spin: -1/2

Example Problem

• Finding quantum numbers for the 3d⁸ electron:
  1. n = 3 (based on level)
  2. l = 2 (d sublevel)
  3. m_l value range: -2 to 2
  4. Ms value: +1/2 or -1/2 (depends on the scenario)
• Determining specific orbitals and spins involves tracking electron placement in orbitals.

Additional Resources

• Links for more examples and practice problems on quantum numbers and electron configurations will be provided in the video description.