Quantum Numbers and Orbitals

Quantum Numbers and Orbitals

Overview of Quantum Numbers

• U_n (Principal Quantum Number)
  • Represents the period in the periodic table.
  • Values: n = 1, 2, 3, …, up to 7.
• L (Azimuthal Quantum Number)
  • Corresponds to orbital types (subshells).
  • Values:
  • L = 0 for S orbitals
  • L = 1 for P orbitals
  • L = 2 for D orbitals
  • L = 3 for F orbitals

Types of Orbitals and Their Characteristics

S Orbitals

• Shape: Spherical
• Characteristics:
  • Only one type of S orbital for each principal quantum number.
  • 1s is the smallest, with each successive S-orbital being larger:
  • 1s < 2s < 3s
  • Analogy: Compared to Gobstoppers, where each layer represents a larger S orbital.
• Electron Capacity: Can hold 2 electrons.
• Existence: All elements have an S orbital, regardless of their position on the periodic table.

P Orbitals

• Shape: Dumbbell
• Characteristics:
  • Only available for L = 1 or higher.
  • There are three degenerate P orbitals:
  • $m_{l} = -1, 0, +1$
  • Each P orbital is orthogonal, existing along the x, y, and z axes.
• Composite View: P orbitals exist simultaneously and have a combined appearance.

D Orbitals

• Characteristics: More complex shapes compared to S and P.
• Number of Orbitals: Five degenerate D orbitals exist.
• Quantum Number: Only present when n = 3 or higher (n cannot be 2 with L = 2).
• Shapes:
  • Four shaped like double dumbbells (or four-leaf clovers).
  • One has a unique shape like a dumbbell with a donut around its center.

F Orbitals

• Characteristics: Even more complex shapes.
• Existence: Only for lanthanides and actinides (n = 4 and n = 6).
• Orientation: Seven orientations exist:
  • From $m<em>{l} = -3$ to $m</em>{l} = +3$.

Quantum Number M Sub L

• Represents the types of orbitals within subshells.
  • S: 1
  • P: 3
  • D: 5
  • F: 7

Quantum Number M Sub S

• Indicates the spin direction of electrons in an orbital.
• Values: +1/2 (spin up) and -1/2 (spin down).
• Importance in Magnetic Resonance Imaging (MRI):
  • Water molecules align in a magnetic field due to the spins.
  • Relaxation of these spins is detected to create images.
  • Reflects the concept of nuclear magnetic resonance (NMR).

Integration of Quantum Numbers

• The combination of n, l, m sub l, and m sub s provides the complete 'address' for an electron:
  • n tells the energy level.
  • l describes the shape of the orbital.
  • m sub l provides specific orbital orientation.
  • m sub s indicates the electron's spin direction.

Example Application of Quantum Numbers

• For the quantum numbers n=3, l=1, m=0, m_s=+rac{1}{2}:
  • The electron is located in the 3p orbital.
  • Energy implies it is at the third energy level in the periodic table, with the P orbital shape, specifically the P_z orbital, indicating an upward spin.

Energetics of Orbitals

• As n increases, distance from the nucleus and energy of electrons increase.
• The electrostatic repulsion between electrons in multiple electron atoms causes subshell splitting, resulting in degeneracy:
  • P Orbitals – 3
  • D Orbitals – 5
  • F Orbitals – 7
• Reinforces that electrons try to achieve maximum separation within their orbitals.