Quantum Mechanics, Wave-Particle Duality, and Electron Configurations

Atomic Emission Spectra:
- Scientists obtained lines of energy emission by exciting atoms (e.g., at an anode).
- Splitting these emission lines into finer spectral lines enabled scientists to deduce and understand the fundamental structure of the atom.
Planck's Quantum Theory:
- Planck postulated that energy does not exist as a continuous stream, but in discrete portions or quanta.
- The fundamental equation for quantum energy is: $E = h \nu$
- $E$ represents energy.
- $h$ represents Planck's constant (referred to as heat or constant).
- $\nu$ represents the frequency of the radiation or light wave.
Einstein's Synthesis and Duality Postulate:
- Einstein applied Planck's quantum energy formula directly to light and equated it to his mass-energy equivalence relation: $h \nu = m c^2$
- $m$ represents the mass of the particle.
- $c$ represents the speed of light.
- Standard numerical values for the speed of light $c$ : $c = 300{,}000\,\text{km/s}$ $c = 3 \times 10^8\,\text{m/s}$
- The term $m$ (mass) represents tangible, material particle properties.
- The term $\nu$ (frequency) represents wave properties derived from Planck's theory.
- By incorporating both mass ( $m$ ) and frequency ($ u$) into a unified description of light, Einstein established the concept of wave-particle duality (double nature of light).

Theoretical Basis and Discovery:
- Einstein received the Nobel Prize for his discovery and theoretical explanation of the photoelectric effect.
- The photoelectric effect confirmed the physical existence of wave-particle duality by showing that light possesses material particle properties.
Mechanism of Electron Ejection:
- A photon is a single quantum or piece of light carrying energy equal to $E = h \nu$ .
- When energy from a light source is directed at a target material composed of atoms, a photon impacts the material.
- The photon imparts sufficient force to overcome internal potential energy obstacles and perform work (the exit work or work function) to liberate an electron.
- Observing the exit/ejection of electronegative particles (electrons) proves that a particle of light (photon) directly influences and ejects a particle of matter (electron).

Extension of Duality to Electrons:
- Following the confirmation of light's dual nature, wave-particle duality was extended to electrons.
- Parameters for identifying properties in quantum formulas:
  - $\lambda$ (wavelength) indicates wave properties.
  - $m$ (mass) indicates particle properties.
  - $h$ (Planck's constant) bridges wave parameters and particle parameters.
Electron Diffraction in Crystal Lattices:
- Diffraction is a wave property characterized by directional bending and alteration of motion upon encountering obstacles.
- Experimental procedure and mechanism:
  - Electrons are directed toward a crystal lattice of a metal ligand.
  - The crystal lattice acts as physical obstacles to the electron stream.
  - Interaction forces between the moving electrons and the crystal lattice cause the electrons to undergo diffraction.
  - After passing the obstacles, diffracted electrons travel to detectors, allowing scientists to confirm that moving electrons display wave properties.
Types of Energy:
- Mathematical instruments calculate energy levels based on potential energy and active properties.
- Light and electrons both exhibit kinetic/wave parameters ( $\lambda$ , $h$ ) and potential/particle parameters ( $m$ ).

Valence Definition:
- Valence represents the number of chemical bonds an atom is capable of forming (chemical combining capacity).
Principal Quantum Number ( $n$ ) and Energy Levels:
- The principal quantum number $n$ identifies the primary shell or energy cell.
- The period number on the periodic table directly specifies the number of occupied energy levels (cells) for an element.
- Subshells fill in designated sequence: $s$ , $p$ , $d$ , etc. (e.g., $2p^6$ , $3s^1$ ).

Steps to Determine Electron Configurations:
- Step 1: Locate the element's period number on the periodic table to establish the highest occupied principal energy level ( $n$ ).
- Step 2: Locate the element's group number to identify the number of valence electrons present in the outermost level.
- Step 3: Begin writing the electron configuration starting from the lowest energy orbitals, where every main energy level starts with an $s$ orbital.
Neutral Sodium ( $\text{Na}$ ) Configuration Example:
- Atomic number = 11 ( $11\text{ protons}$ , $11\text{ electrons}$ ).
- Period number = 3 (outermost level is $n = 3$ ).
- Group number = 1 (1 valence electron in outer shell).
- Complete electron configuration for neutral Sodium: $1s^2 2s^2 2p^6 3s^1$
- Electron sum verification: $2 + 2 + 6 + 1 = 11\text{ electrons}$

Question: How is the complete electron configuration derived for a Sodium cation ( $\text{Na}^+$ )?
- Response:
  - A charge of $+1$ indicates a positively charged cation with 1 fewer electron than the neutral state.
  - Total electrons for $\text{Na}^+$ : $11 - 1 = 10\text{ electrons}$
  - Step-by-step orbital filling:
    1. $1s^2$ accounts for $2\text{ electrons}$ (total = 2).
    2. $2s^2$ accounts for $2\text{ electrons}$ (total = 4).
    3. $2p^6$ accounts for $6\text{ electrons}$ (total = 10).
  - The complete configuration for $\text{Na}^+$ is $1s^2 2s^2 2p^6$ .
Question: What is the active valence level for the sodium cation ( $\text{Na}^+$ )?
- Response:
  - When neutral sodium loses its single valence electron from the $3s$ orbital, the $3s$ subshell becomes completely empty ( $3s^0$ ).
  - Because the third level is empty, the outer valence level reverts to the entire second shell.
  - The complete valence shell for $\text{Na}^+$ is: $2s^2 2p^6$
Session Break:
- A 10-minute lecture break follows these discussions.