Comprehensive Study Guide: Electron Configurations, Orbital Diagrams, and Quantum Exceptions

Course Announcements and Classroom Guidelines

• Lecture Conduct and Noise Policy:

  • Side-of-room talking during lectures has raised complaints over the last three lectures.
  • Because of hearing impairment near the front of the room, whispering is mandatory if conversing during lecture.
  • Normal talking voices are strictly prohibited while lecture material is being presented.
  • Collaborative dialogue at normal volume is encouraged when working actively on class problems.

• Academic Performance and Support:

  • Class academic performance is strong grade-wise; maintaining high effort is essential.
  • Yellow periodic table reference sheets are available in lab (dozens available upon request).

• Instructional Method Shift:

  • Classroom board visibility issues (gray background design) necessitate using a 1990s-style overhead projector for writing out examples.

• Schedule and Upcoming Activities:

  • No class on Monday due to Labor Day.
  • Naming-focused study games will be introduced following Labor Day.
  • Mastery of electron configurations requires completing $10$ to $20$ practice problems.

Fundamentals of Electron Configuration

• Definition and Purpose:

  • An electron configuration is an organizational system used to locate and describe the distribution of subatomic electrons within an atom.
  • It identifies subshell regions and energy levels where electrons are most likely to be found.

• Comparison of Atomic Models:

  • Bohr Model: Represents the atom as a central nucleus containing protons and neutrons, surrounded by two-dimensional ($2D$) concentric circular orbits representing energy levels ($n$).
  • Quantum Mechanical Model: Expands spatial organization into three dimensions ($3D$) using subshells ($s$, $p$, $d$, $f$) mapped directly via the periodic table.

Quantum Numbers and Subshell Classification

• Principal Quantum Number ($n$):

  • Represents the main energy level of an electron.
  • Corresponds directly to the periods (rows $1$ through $7$) down the periodic table.

• Subshell Types, Shapes, and Capacities:

  • $s$ Subshell:
  • Location: First two columns (Groups 1 and 2) of the periodic table.
  • Shape: Spherical / circle.
  • Orbitals: $1$ orbital box.
  • Maximum Capacity: $2$ electrons.
  • $p$ Subshell:
  • Location: Right side of the periodic table (Groups 13 through 18).
  • Shape: Dumbbell / pair of lobes / infinity symbol shape ($\infty$).
  • Orbitals: $3$ orbital boxes.
  • Maximum Capacity: $6$ electrons.
  • $d$ Subshell:
  • Location: Middle section of the periodic table (transition metals).
  • Shape: Double dumbbell / four lobes (resembling an $X$ shape or pacifier).
  • Numbering Rule: Encoded as $(n - 1)$. Row $4$ transition metals fill the $3d$ subshell (e.g., $4s$ followed by $3d$).
  • Orbitals: $5$ orbital boxes.
  • Maximum Capacity: $10$ electrons.
  • $f$ Subshell:
  • Location: Bottom two extracted rows (lanthanides and actinides).
  • Shape: Complex multi-lobed structures.
  • Orbitals: $7$ orbital boxes.
  • Maximum Capacity: $14$ electrons.

• Subshell Progression and Theoretical Extensions:

  • Number of orbitals per subshell follows odd integer values: $1, 3, 5, 7$.
  • Subshell availability by energy level:
  • Level $1$: $s$
  • Level $2$: $s, p$
  • Level $3$: $s, p, d$
  • Level $4$: $s, p, d, f$
  • Level $5$: Theoretically permits a $g$ subshell ($9$ orbitals, $18$ electrons), followed by $h$ and $i$ at higher levels.

• Reading the Periodic Table:

  • Read configurations like standard text: top-left to right, row by row.
  • Helium ($He$) is treated as part of the $1s$ subshell, filling it as $1s^2$.

Detailed Examples of Complete Electron Configurations

• Magnesium ($Mg$, Atomic Number $12$):

  • Sequence: Row 1 ($1s^2$) $\rightarrow$ Row 2 ($2s^2, 2p^6$) $\rightarrow$ Row 3 ($3s^2$)
  • Complete Configuration: $1s^2 2s^2 2p^6 3s^2$

• Phosphorus ($P$, Atomic Number $15$):

  • Sequence: Row 1 ($1s^2$) $\rightarrow$ Row 2 ($2s^2, 2p^6$) $\rightarrow$ Row 3 ($3s^2, 3p^3$)
  • Complete Configuration: $1s^2 2s^2 2p^6 3s^2 3p^3$

• Scandium ($Sc$, Atomic Number $21$):

  • Sequence: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1$
  • Note: Transition into $3d$ occurs after $4s$.

• Ruthenium ($Ru$, Atomic Number $44$):

  • Sequence: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^6$

Rules for Orbital Diagrams

• Visual Representation:

  • Orbitals are drawn as boxes; electrons are drawn as vertical arrows.
  • Opposite arrow directions (up $\uparrow$ and down $\downarrow$) denote opposite electron spins.
  • Electrons occupy distinct spatial regions due to electrostatic repulsion.

• Hund's Rule:

  • When filling degenerate orbitals (orbitals of equal energy within the same subshell), place one spin-up electron in each orbital box before pairing any electrons with spin-down arrows.
  • Analogy: Roommates occupying separate bedrooms before sharing bedrooms.

• Energy Progression in Diagrams:

  • Higher energy levels correspond to greater physical distance from the nucleus.
  • Outer electrons experience shielding/repulsion from inner core electrons.

• Orbital Diagram for Phosphorus ($P$):

  • $1s$ orbital box: [$\uparrow\downarrow$]
  • $2s$ orbital box: [$\uparrow\downarrow$]
  • $2p$ orbital boxes: [$\uparrow\downarrow$][$\uparrow\downarrow$][$\uparrow\downarrow$]
  • $3s$ orbital box: [$\uparrow\downarrow$]
  • $3p$ orbital boxes: [$\uparrow$][$\uparrow$][$\uparrow$]

Alternative Order Determination: Aufbau Diagonal Method

• Grid Setup for Diagonal Reading:

  • Write subshells in vertical columns:
  • Column 1: $1s, 2s, 3s, 4s, 5s, 6s, 7s$
  • Column 2: $2p, 3p, 4p, 5p, 6p, 7p$
  • Column 3: $3d, 4d, 5d, 6d$
  • Column 4: $4f, 5f$
  • Draw parallel diagonal arrows down and to the left (from tail to head).

• Filling Order Derived from Diagonals:

  • $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow 4d \dots$

Valence and Core Electron Determination

• Definitions:

  • Valence Electrons: Electrons in the outermost principal quantum level (highest $n$ value), restricted to $s$ and $p$ subshells for main group calculations. These electrons drive chemical reactivity and ionic charge formation.
  • Core Electrons: Inner electrons occupying completed lower quantum shells close to the nucleus. They do not participate in standard chemical reactions.

• Mathematical Calculation:

  • $\text{Core Electrons} = \text{Total Electrons} - \text{Valence Electrons}$

• Worked Examples:

  • Oxygen ($O$, Atomic Number $8$):
  • Configuration: $1s^2 2s^2 2p^4$
  • Highest $n$: $2$ ($2s^2 2p^4$
  • Valence Electrons: $2 + 4 = 6$
  • Core Electrons: $8 - 6 = 2$
  • Phosphorus ($P$, Atomic Number $15$):
  • Configuration: $1s^2 2s^2 2p^6 3s^2 3p^3$
  • Highest $n$: $3$ ($3s^2 3p^3$
  • Valence Electrons: $2 + 3 = 5$
  • Core Electrons: $15 - 5 = 10$
  • Neon ($Ne$, Atomic Number $10$):
  • Configuration: $1s^2 2s^2 2p^6$
  • Highest $n$: $2$ ($2s^2 2p^6$
  • Valence Electrons: $2 + 6 = 8$ (complete octet, chemically inert)
  • Core Electrons: $10 - 8 = 2$
  • Scandium ($Sc$, Atomic Number $21$):
  • Configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1$
  • Highest $n$: $4$ ($4s^2$
  • Valence Electrons: $2$
  • Core Electrons: $21 - 2 = 19$
  • Ruthenium ($Ru$, Atomic Number $44$):
  • Configuration: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^2 4d^6$
  • Highest $n$: $5$ ($5s^2$
  • Valence Electrons: $2$
  • Core Electrons: $44 - 2 = 42$

Complete vs. Noble Gas Abbreviated Configurations

• Methodology for Abbreviated Configurations:

  1. Locate the target element on the periodic table.
  2. Move up one row to the noble gas at the end of that row (Group 18).
  3. Write the noble gas symbol inside square brackets $[\text{Noble Gas}]$.
  4. Write the remaining configuration for the valence electrons in the current row.

• Comparative Examples:

  • Magnesium ($Mg$):
  • Complete: $1s^2 2s^2 2p^6 3s^2$
  • Preceding Noble Gas: Neon ($Ne$
  • Abbreviated: [Ne]\n,3s^2
  • Scandium ($Sc$):
  • Complete: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1$
  • Preceding Noble Gas: Argon ($Ar$
  • Abbreviated: [Ar]\n,4s^2 3d^1
  • Ruthenium ($Ru$):
  • Preceding Noble Gas: Krypton ($Kr$
  • Abbreviated: [Kr]\n,5s^2 4d^6

• Excited State Configurations:

  • Denoted by an asterisk ($X^*$).
  • Occurs when an electron absorbs energy and is promoted to a higher energy subshell out of normal order.
  • Relaxes back to ground state by emitting light.

Electron Configurations of Ions

• Cation Removal Rule (Crucial Exception):

  • When forming positive ions (cations), electrons are ALWAYS removed from the highest principal quantum number ($n$) shell first (the valence $s$ electrons), BEFORE removing electrons from lower $n$ subshells like $d$

• Ion Configuration Practice:

  • Carbon ($C$) vs. Carbon Cation ($C^{2+}$):
  • Neutral $C$ ($6 e^-$): $1s^2 2s^2 2p^2$
  • $C^{2+}$ ($4 e^-$): Remove $2$ electrons from highest subshell ($2p$) $\rightarrow$ $1s^2 2s^2$
  • Oxygen ($O$) vs. Oxide Anion ($O^{2-}$):
  • Neutral $O$ ($8 e^-$): $1s^2 2s^2 2p^4$
  • $O^{2-}$ ($10 e^-$): Add $2$ electrons to valence shell $\rightarrow$ $1s^2 2s^2 2p^6$ (isoelectronic with Neon)
  • Iron Cation ($Fe^{2+}$):
  • Neutral $Fe$ ($26 e^-$): [Ar]\n,4s^2 3d^6
  • $Fe^{2+}$ ($24 e^-$): Remove $2$ electrons from highest $n$ ($4s$) $\rightarrow$ [Ar]\n,3d^6
  • Cobalt Cation ($Co^{3+}$):
  • Neutral $Co$ ($27 e^-$): [Ar]\n,4s^2 3d^7
  • $Co^{3+}$ ($24 e^-$): Remove $2$ electrons from $4s$ and $1$ electron from $3d$ $\rightarrow$ [Ar]\n,3d^6
  • Note: $Fe^{2+}$ and $Co^{3+}$ are isoelectronic ($3d^6$).

Anomalous Electron Configurations (Exceptions)

• Mechanism of Anomalies:

  • The energy gap between the $ns$ and $(n-1)d$ subshells is extremely small.
  • Subshells gain thermodynamic stability when they are exactly half-filled ($d^5$) or completely filled ($d^{10}$).
  • An electron shifts from the filled $s$ orbital to the $d$ subshell to achieve this stability.

• Chromium Family Exception ($s^1 d^5$):

  • Chromium ($Cr$, Atomic Number $24$):
  • Expected: [Ar]\n,4s^2 3d^4
  • Actual: [Ar]\n,4s^1 3d^5
  • Molybdenum ($Mo$, Atomic Number $42$):
  • Follows the same shift: [Kr]\n,5s^1 4d^5

• Copper Family Exception ($s^1 d^{10}$):

  • Copper ($Cu$, Atomic Number $29$):
  • Expected: [Ar]\n,4s^2 3d^9
  • Actual: [Ar]\n,4s^1 3d^{10}
  • Silver ($Ag$) and Gold ($Au$):
  • Follow the same shift: $s^1 d^{10}$

Questions & Classroom Discussion

• Question: How many orbitals are drawn for $p$, $d$, and $f$ subshells in an orbital diagram?

  • Answer: $p$ has $3$ orbital boxes, $d$ has $5$ orbital boxes, and $f$ has $7$ orbital boxes.

• Question: Are students required to draw the 3D spatial shapes of $d$ and $f$ orbitals on exams?

  • Answer: No. $f$ orbitals are never tested visually. $d$ orbitals (four lobes/pacifier shape) only appear in multiple-choice identification questions.

• Question: Why do cations lose $4s$ electrons before $3d$ electrons when forming transition metal ions?

  • Answer: The $4s$ orbital has a higher principal quantum number ($n = 4$) than $3d$ ($n = 3$), making $4s$ the valence shell. Valence electrons are furthest from the nucleus and most accessible for removal during ionization.