Electrons in Atoms and the Periodic Table

Quantum Mechanics

• Schrodinger’s wave equation:
  • Incorporates both wave-like and particle-like behavior of electrons.
  • Represented as a lowercase Greek psi ((\psi)).
  • The square of the wave function ((\psi^2)) provides a probability density map for electron location.

Quantum Numbers

• Solving the wave equation yields orbitals described by three quantum numbers:
  • Principal Quantum Number (n): Indicates energy level and is a positive integer ( n \geq 1 ).
  • Angular Momentum Quantum Number (l): Defines orbital shape, ranging from 0 to (n - 1).
  • Magnetic Quantum Number (m_l): Describes orbital orientation.

Quantum Mechanical Model

• Electrons do not follow precise paths around the nucleus but are located in orbitals defined by quantum numbers:
  • Terminology: Shell, subshell, atomic orbital

Principal Quantum Number (n)

• Describes the energy level and shell:
  • ( n = 1 ) corresponds to the lowest energy state (1s orbital).
  • Higher (n) values signify higher energy orbitals (e.g., ( n = 2, 3, …)).

Energy with Principal Quantum Number

• Energy increases with higher ( n ):
  • ( n = 1 ): 1s has lowest energy.
  • Orbitals filled in order of increasing energy levels.

Angular Momentum Quantum Number (l)

• Defines the shape of the orbital:
  • Values: 0 (s), 1 (p), 2 (d), 3 (f)
  • Total subshells equal to ( n ) (e.g. ( n = 2 ) contains s and p subshells).

Types of Orbitals

1. s Orbitals:
   • Spherical shape,
   • Size increases with ( n ).
   • Example: 2s larger than 1s.
2. p Orbitals:
   • Starts at ( n = 2 ), dumbbell-shaped,
   • Three orbitals: px, py, pz.
3. d Orbitals:
   • Starts at ( n = 3 ), contains five orbitals,
   • Four d orbitals with four lobes and one resembling a p orbital with a doughnut.

Learning Checks on Orbitals

• Different types and numbers of orbitals for each principal quantum number:
  1. For ( n = 1 ): 1s (1 orbital)
  2. For ( n = 2 ): 2s, 2p (4 orbitals)
  3. For ( n = 3 ): 3s, 3p, 3d (9 orbitals)
  4. For ( n = 4 ): 4s, 4p, 4d, 4f (16 orbitals).

Pauli Exclusion Principle

• No two electrons in an atom can have identical quantum numbers; they must have opposite spins.

Energy Levels of Orbitals

• In one-electron systems (like hydrogen), all orbitals on the same energy level are degenerate (same energy).
• In multi-electron atoms, energy levels differ due to electron-electron interactions. Calculate energy using (n+l) rule.

Electronic Configuration

• Distribution of electrons across orbitals:
  • Ground state configuration minimizes energy.
  • Abbreviated configurations use previous noble gas configurations.
• Example: (4p^5) indicates:
  • 4: Principal quantum number,
  • p: Type of subshell,
  • 5: Number of electrons.

Aufbau Principle

• Electrons fill the lowest energy orbitals first:
  • Sequence follows increasing atomic number, adding protons and electrons sequentially.
  • Order of filling is crucial for predicting electron configurations.

Periodic Table Application

• Orbitals fill in conjunction with periods and groups.
• Elements in the same group have similar outer shell configurations affecting their chemical reactivity.

General Configuration Rules

• Electrons occupy the lowest energy orbitals first, with maximum two electrons per orbital without pairing unless necessary. Orbital filling order:
  • s = 2, p = 6, d = 10, f = 14.

Condensed Electron Configuration

• Utilizes noble gas core to simplify electron notation:
  • Core electrons vs. valence electrons key in chemical bonding.

Valence vs. Core Electrons

• Valence electrons involved in bonding, whereas core electrons fill inner shells.
• Example: Silicon (4 valence, 10 core) and Selenium (6 valence, others core).