Quantum Mechanical Model and Electron Configurations Note

Quantum Mechanical Model of the Atom

• Dependent Quantities and Measurement Uncertainty

  • In quantum mechanics, certain physical quantities are dependent on one another rather than independent; measuring one dependent quantity fundamentally alters or destroys the ability to simultaneously measure the other.

• Wave Functions and Probability Distributions

  • Quantum mechanics describes electrons by their energy (the energy associated with their wave function) and their probable spatial locations.
  • Wave functions ($   \psi   $) are solutions to complex mathematical equations in three-dimensional space that generate probability distributions for finding an electron in a given region of space.
  • Electron behavior mirrors light waves, possessing associated electric fields that function similarly to electromagnetic light waves.

• Comparison Between the Bohr Model and Quantum Mechanics

  • The Bohr model assumes fixed circular or spherical orbits around the nucleus.
  • The quantum mechanical model replaces fixed orbits with complex 3D orbital probability regions derived from wave functions.
  • On an energy scale, the models share similarities: energy levels correlate with literal distance from the atomic nucleus.

Principal Quantum Numbers and Energy Levels

• Principal Quantum Number ($n$)

  • Defined as an integer ($n = 1, 2, 3, 4, \dots$) that specifies the main energy level of an electron and its relative distance from the nucleus.

• Distance and Energy Relationship

  • The energy required to hold an electron in place is inversely proportional to the square of its distance from the nucleus ($E \propto \frac{1}{r^2}$).
  • Lower energy levels ($n = 1$) are closest to the core of the atom (the nucleus), where electrons are held most tightly with lower potential energy.
  • Higher energy levels ($n = 2, 3, 4, \dots$) are positioned progressively farther from the core, requiring greater energy to maintain electrons at those distances.

• Circumference and Electron Capacity per Level

  • Lower energy levels have a smaller physical circumference around the nucleus, limiting the number of electrons they can accommodate.
  • Higher energy levels have longer circumferences, enabling them to accommodate larger numbers of electrons.
  • Electron capacity sequence by principal energy level:
  • Level $n = 1$: Maximum of $2\,\text{electrons}$
  • Level $n = 2$: Maximum of $8\,\text{electrons}$
  • Level $n = 3$: Maximum of $18\,\text{electrons}$
  • Level $n = 4$: Maximum of $32\,\text{electrons}$
  • Energy levels $n = 4, 5, 6, 7$ cap out at $32\,\text{electrons}$ because atoms exceeding these dimensions become energetically unfavorable and structurally unstable, causing them to fall apart.

Sublevels, Orbital Shapes, and Spatial Geometries

• Sublevels and Orbital Classifications

  • Each principal energy level consists of specific energy sublevels designated as $s$, $p$, $d$, and $f$.
  • Smaller atoms contain only $s$ and $p$ orbitals.

• Orbital Quantities and Electron Capacities per Sublevel

  • $s$ sublevel: Contains $1\,\text{orbital}$ (holds up to $2\,\text{electrons}$)
  • $p$ sublevel: Contains $3\,\text{orbitals}$ (holds up to $6\,\text{electrons}$)
  • $d$ sublevel: Contains $5\,\text{orbitals}$ (holds up to $10\,\text{electrons}$)
  • $f$ sublevel: Contains $7\,\text{orbitals}$ (holds up to $14\,\text{electrons}$)

• Spatial Geometries of Orbitals

  • $s$ Orbitals:
  • Spherical in shape, arranged like concentric shells around the nucleus.
  • $1s$ is closest to the nucleus; $2s$ is physically larger and farther from the positively charged nucleus, requiring higher energy.
  • Energy level $n = 1$ contains exclusively an $s$ orbital because the physical distance is too small to accommodate more complex geometries.
  • $p$ Orbitals:
  • Orient along three spatial dimensions ($x, y, z$).
  • First appear at energy level $n = 2$, yielding a combined level total of $8\,\text{electrons}$ ($1s$ orbital + $3p$ orbitals = $2 + 6 = 8\,\text{electrons}$).
  • $d$ Orbitals:
  • Arise from spatial interactions across two of the three dimensions, yielding $5$ spatial orientation combinations.
  • First appear at energy level $n = 3$.
  • Possess complex geometries, including the uniquely shaped $z^2$ orbital.
  • Level $n = 3$ total capacity: $1\,\text{s orbital} (2\,e^-) + 3\,\text{p orbitals} (6\,e^-) + 5\,\text{d orbitals} (10\,e^-) = 18\,\text{electrons}$.
  • $f$ Orbitals:
  • Contain $7$ distinct orbital types.
  • First appear at energy level $n = 4$.
  • Possess highly complex geometries and are relevant primarily in large elements with high electron counts.

Electron Spin and Orbital Representation

• Fundamental Spin States

  • Electrons act as microscopic magnetic dipoles with identical charge signs that inherently repel one another.
  • Spin is a binary fundamental quantum property existing in two states: spin-up and spin-down.
  • Two electrons occupying the same orbital in identical spin states (e.g., spin-up / spin-up) create an energetically unfavorable, higher-energy configuration due to repulsion.
  • Placing electrons in opposite spin states (spin-up / spin-down) minimizes magnetic repulsion, creating a lower-energy, stable system.

• Shorthand Notation for Electrons

  • Single electrons are represented in orbital diagrams using half-arrows ($   \upharpoonleft   $ or $   \downharpoonright   $).
  • Arrow direction indicates spin state (upward = spin-up, downward = spin-down).
  • The symbol $e^-$ serves as standard shorthand for an electron.
  • Each individual orbital box or line can hold a maximum of $2\,\text{electrons}$ of opposite spin.

Orbital Energy Hierarchy and Filling Rules

• Aufbau Principle (Energy Ordering)

  • Electrons occupy orbitals in strict order of increasing energy, not in numerical sequence of principal quantum numbers.
  • Lowest energy orbitals fill completely before higher energy orbitals begin populating.
  • Energy hierarchy of sublevels:
  • $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f \dots$
  • Critical Energy Inversions:
  • The $4s$ orbital is lower in energy than the $3d$ orbitals, so $4s$ fills completely before $3d$ begins filling.
  • The $6s$ orbital is lower in energy than the $4f$ orbitals and fills before $4f$, despite being two principal energy levels apart.

• Degenerate Orbitals

  • Orbitals within the same sublevel that possess identical energy levels are termed degenerate.
  • The three $2p$ orbitals are degenerate with respect to each other.
  • The five $3d$ orbitals are degenerate with respect to each other.
  • The seven $4f$ orbitals are degenerate with respect to each other.

• Hund's Rule

  • When filling degenerate orbitals, electrons enter empty orbitals individually with parallel spins before pairing up.
  • Single occupation of degenerate orbitals minimizes electron-electron repulsion, making it energetically favorable over immediate pairing.

Writing Electron Configurations

• Standard Notation Structure

  • Written using the format: Principal Energy Level ($n$) + Sublevel Letter ($s, p, d, f$) + Superscript Electron Count.

• Step-by-Step Configuration Examples

  • Hydrogen ($H$, atomic number 1): $1\,\text{electron}$
  • \n    1s^1\n    
  • Helium ($He$, atomic number 2): $2\,\text{electrons}$
  • \n    1s^2\n    
  • Boron ($B$, atomic number 5): $5\,\text{electrons}$
  • \n    1s^2\,2s^2\,2p^1\n    
  • Nitrogen ($N$, atomic number 7): $7\,\text{electrons}$
  • Applies Hund's rule across three degenerate $2p$ orbitals:
  • \n    1s^2\,2s^2\,2p^3\n    
  • Oxygen ($O$, atomic number 8): $8\,\text{electrons}$
  • The 4th $2p$ electron pairs up in the first $2p$ orbital:
  • \n    1s^2\,2s^2\,2p^4\n    
  • Silicon ($Si$, atomic number 14): $14\,\text{protons}, 14\,\text{electrons}$
  • Sequential fill: $1s\,(2e^-) \rightarrow 2s\,(2e^-) \rightarrow 2p\,(6e^-) \rightarrow 3s\,(2e^-) \rightarrow 3p\,(2e^-)$
  • Full configuration: \n    1s^2\,2s^2\,2p^6\,3s^2\,3p^2\n    
  • The two $3p$ electrons occupy separate degenerate orbitals with parallel spins.
  • Chlorine ($Cl$, atomic number 17): $17\,\text{electrons}$
  • Full configuration: \n    1s^2\,2s^2\,2p^6\,3s^2\,3p^5\n    
  • Titanium ($Ti$, atomic number 22): $22\,\text{electrons}$
  • Full configuration: \n    1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^2\n    
  • Selenium ($Se$, atomic number 34): $34\,\text{electrons}$
  • Full configuration: \n    1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^{10}\,4p^4\n    

Condensed Noble Gas Notation and Core vs. Valence Electrons

• Distinction Between Electron Categories

  • Inner (Core) Electrons: Electrons belonging to completely filled interior energy levels, equivalent to the preceding noble gas structure.
  • Outer Electrons: All electrons situated outside the noble gas core, occupying the highest principal energy level plus unfilled/filled $d$ or $f$ sublevels.
  • Valence Electrons: Electrons located strictly in the highest principal quantum number level ($n$). These dictate chemical reactivity and bonding.
  • Note on $d$ electrons: Although $3d$ is higher in energy than $4s$, $3d$ electrons are not valence electrons because $n=3$ is not the highest principal energy level.

• Condensed Noble Gas Shorthand

  • Replaces core electrons with the bracketed symbol of the preceding noble gas ($[He]$, $[Ne]$, $[Ar]$, $[Kr]$).
  • Noble gases have completely filled valence shells ($s^2\,p^6$), making them chemically unreactive and stable.
  • Sodium ($Na$, atomic number 11):
  • Uncondensed: \n    1s^2\,2s^2\,2p^6\,3s^1\n    
  • Condensed: \n    [Ne]\,3s^1\n    
  • Phosphorus ($P$, atomic number 15):
  • Condensed: \n    [Ne]\,3s^2\,3p^3\n    
  • Argon ($Ar$, atomic number 18):
  • Condensed: \n    [Ne]\,3s^2\,3p^6\n    
  • Titanium ($Ti$, atomic number 22):
  • Condensed: \n    [Ar]\,4s^2\,3d^2\n    
  • Selenium ($Se$, atomic number 34):
  • Condensed: \n    [Ar]\,4s^2\,3d^{10}\,4p^4\n    

Octet Rule, Ion Formation, and Isoelectronicity

• The Octet Rule

  • Observation that atoms gain, lose, or share electrons to attain an $s^2\,p^6$ outer valence configuration containing $8\,\text{electrons}$, matching the stability of noble gases.

• Formation of Cations and Anions

  • Sodium Cation ($Na^+$):
  • Neutral $Na$ ($11\,\text{protons}, 11\,\text{electrons}$): \n    [Ne]\,3s^1\n    
  • Discards its single $3s^1$ valence electron to form $Na^+$ ($11\,\text{protons}, 10\,\text{electrons}$).
  • Resulting configuration: \n    1s^2\,2s^2\,2p^6\n     (or $[Ne]$), fulfilling the octet rule with $8\,\text{valence electrons}$ in $n = 2$
  • Oxide Anion ($O^{2-}$):
  • Neutral $O$ ($8\,\text{protons}, 8\,\text{electrons}$): \n    1s^2\,2s^2\,2p^4\n    
  • Gains $2\,\text{electrons}$ to form $O^{2-}$ ($8\,\text{protons}, 10\,\text{electrons}$).
  • Resulting configuration: \n    1s^2\,2s^2\,2p^6\n     (or $[Ne]$).

• Isoelectronic Species

  • Definition: Chemical species (atoms or ions) that possess identical total electron counts and identical ground-state electron configurations.
  • Example: $O^{2-}$, $Na^+$, and neutral $Ne$ all contain $10\,\text{electrons}$ with configuration \n  1s^2\,2s^2\,2p^6\n  , making them mutually isoelectronic.

• Group Ionization Trends

  • Group 1A (Alkali Metals): Possess an $s^1$ valence configuration; readily lose $1\,\text{electron}$ to form $+1$ cations isoelectronic with the preceding noble gas.
  • Group 7A / Group 17 (Halogens): Possess a $p^5$ valence configuration; readily gain $1\,\text{electron}$ to form $-1$ anions (e.g., chloride $Cl^-$, bromide $Br^-$) isoelectronic with the succeeding noble gas.

Periodic Table Structure and Sublevel Blocks

• Structural Correlative Features

  • Rows (Periods): Directly correspond to the highest occupied principal energy level ($n$) for valence electrons.
  • Period 1 = Energy level $n = 1$
  • Period 2 = Energy level $n = 2$
  • Period 5 = Energy level $n = 5$ (e.g., Xenon, $Xe$, atomic number 54, has valence configuration \n    5s^2\,5p^6\n    )
  • Columns (Groups): Indicate the specific valence electron configuration and count.

• Sublevel Blocks on the Periodic Table

  • $s$ Block: First two columns on the left (Group 1A and 2A), corresponding to $s^1$ and $s^2$ configurations.
  • $p$ Block: Rightmost six columns (Groups 3A through 8A / Groups 13 through 18), corresponding to $p^1$ through $p^6$ configurations.
  • $d$ Block: Middle transition metal section, corresponding to filling $d$ orbitals ($d^1$ to $d^{10}$). For example, Iron ($Fe$, atomic number 26) has outer configuration \n  4s^2\,3d^6\n  
  • $f$ Block: Bottom isolated two rows (lanthanides and actinides), corresponding to filling $f$ orbitals ($f^1$ to $f^{14}$). Contains elements like Lanthanum ($La$) and Lutetium ($Lu$), which are rarely encountered in introductory chemistry.