Study Notes on Quantum Numbers and Electron Configuration

Quantum Numbers and Orbitals

• Quantum Numbers Overview
  • Four types of quantum numbers describe the properties of electrons in atomic orbitals:
  1. Principal Quantum Number ($n$)
  2. Angular Momentum Quantum Number ($l$)
  3. Magnetic Quantum Number ($m_l$)
  4. Spin Quantum Number ($m_s$)

Principal Quantum Number ($n$)

• Defines the energy level of the electron and the size of the orbital.
• Integer values: $n = 1, 2, 3, …$.

Angular Momentum Quantum Number ($l$)

• Defines the shape of the orbital.
• Values based on the principal quantum number:
  • $l = 0$ for s orbitals
  • $l = 1$ for p orbitals
  • $l = 2$ for d orbitals
  • $l = 3$ for f orbitals

Magnetic Quantum Number ($m_l$)

• Defines the orientation of the orbital in space.
• For each $l$, $m_l$ can take values from $-l$ to $+l$:
  • s orbitals ($l = 0$): only one orientation, $m_l = 0$.
  • p orbitals ($l = 1$): three orientations, $m_l = -1, 0, +1$.
  • d orbitals ($l = 2$): five orientations, $m_l = -2, -1, 0, +1, +2$.
  • f orbitals ($l = 3$): seven orientations, $m_l = -3, -2, -1, 0, +1, +2, +3$.

Spin Quantum Number ($m_s$)

• Represents the spin of the electron.
  • Two possible values: $+\frac{1}{2}$ (spin up) and $-\frac{1}{2}$ (spin down).

Orbital Representations

• Electrons in orbitals are represented by box diagrams.
• Each box corresponds to one orbital defined by the quantum numbers.

Filling Orbitals

• Electrons fill orbitals in order of increasing energy (Aufbau principle).
• The first energy level ($n = 1$) can hold up to 2 electrons, corresponding to the 1s orbital.
• The second energy level ($n = 2$) can hold up to 8 electrons (2 in 2s, 6 in 2p).
• The third energy level ($n = 3$) can hold 18 electrons (2 in 3s, 6 in 3p, 10 in 3d).
• The fourth energy level ($n = 4$) can hold 32 electrons (2 in 4s, 6 in 4p, 10 in 4d, 14 in 4f).

Orbital Diagrams for Elements

• The arrangement of electrons in an atom is described by the electron configuration and orbital diagrams.

Example: Helium (Atomic Number 2)

• Electron Configuration:
  • 1s$^2$
  • Representation: Two arrows pointing in opposite directions in the box for 1s.

Example: Carbon (Atomic Number 6)

• Electron Configuration:
  • 1s$^2$, 2s$^2$, 2p$^2$
  • Orbital diagram indicates distribution of electrons with consideration of Hund's rule - maximizing unpaired spins in degenerate orbitals before pairing.

Hund's Rule

• States that electrons must occupy every orbital singly before any orbital is doubly occupied in the same subshell.

Pauli Exclusion Principle

• No two electrons in an atom can have the same set of four quantum numbers.

Examples of Electron Configuration

• Lithium (Atomic Number 3):
  • Electron Configuration: 1s$^2$, 2s$^1$
• Boron (Atomic Number 5):
  • Electron Configuration: 1s$^2$, 2s$^2$, 2p$^1$
• Neon (Atomic Number 10):
  • Electron Configuration: 1s$^2$, 2s$^2$, 2p$^6$
  • Valence Electrons: 8 (fully filled outer shell, stable configuration).

Isoelectronic Species

• Species that have the same number of electrons and same electron configuration, e.g., Ne, F$^{-}$, Na$^{+}$.

Transition Metals

• Transition metals fill their d orbitals, with sometimes complicated electron configurations due to energy level shifts.
  • Example: Chromium (24 electrons): 4s$^1$ 3d$^5$ due to stability from half-filled d orbitals.
• Example: Copper (29 electrons): 4s$^1$ 3d$^{10}$ for similar stability reasons.

Magnetic Properties

• Diamagnetic: All electrons paired; do not respond to magnetic fields.
• Paramagnetic: Unpaired electrons present; attracted to magnetic fields.

Conclusion

• Understanding electron configurations and orbital diagrams is essential for predicting the properties and behaviors of elements in chemistry.
• Review the filling order, Hund's rule, and Pauli's exclusion principle for a solid grasp of these foundational concepts in atomic structure.