Electron Configurations, Orbital Shapes, and Quantum Mechanical Rules

Orbital Characteristics and Quantum Nodes

• $s$ Orbital Spatial Distributions and Nodes:

  • $s$ subshells are represented spatially as concentric spherical regions around the atomic nucleus.

  • The number of nodes in an $s$ orbital increases as the principal quantum number ($n$) increases:

    • A $6s$ orbital contains $5$ nodes.

    • A $7s$ orbital contains $6$ nodes.

  • Mathematically, the number of radial nodes in an $s$ orbital equals $n - 1$.

• Limits of Known Elements and Theoretical $8s$ Orbitals:

  • Currently, the $118$ known elements in the periodic table fill orbitals up to the $7s$ subshell.

  • An $8s$ orbital does not exist for ground-state elements in standard conditions.

  • Theoretical existence of an $8s$ orbital could occur under extreme physical conditions, such as exceptionally high pressure, high temperature, or novel nuclear reactions that synthesize new superheavy elements far from the nuclear center.

• Physical Meaning of Quantum Nodes:

  • A node is a region or surface where the probability of finding an electron is strictly zero ($P = 0$).

  • Nodes act as strict forbidden zones for electron density.

  • Analogy: A node is like a dedicated bicycle lane that non-bicycle vehicles (cars) are strictly forbidden from entering.

• Comparison Between Bohr Model and Quantum Mechanical Model:

  • Bohr Theory: Models electrons as particles moving along simple, fixed circular orbits.

  • Quantum Mechanical Model: Models orbitals as three-dimensional probability distributions or spatial zones within which electrons travel around the nuclear center.

  • Size Relationship: $1s < 2s < 3s$. As the principal quantum number ($n$) increases, the orbital size expands, meaning electrons are located farther away from the nuclear center.

Quantum Number Rules Governing Orbital Existence and Shapes

• $p$ Orbital Characteristics and Orientations:

  • $p$ orbitals possess a dumbbell shape representing the spatial region where electrons travel.

  • The three spatial orientations of $p$ orbitals lie along the Cartesian axes: $p_x$, $p_y$, and $p_z$.

  • The spatial orientation of an orbital is strictly determined by the magnetic quantum number ($m_l$).

  • Orbital size scales with principal quantum number ($n$): a $2p$ orbital is smaller in spatial extent than a $3p$ orbital.

• Mathematical Proof for the Non-Existence of $1p$ Orbitals:

  • Quantum number rules dictate that the angular momentum quantum number ($l$) can take integer values from $0$ to $n - 1$:         l \nin \{0, 1, 2, \dots, n - 1\}

  • The magnetic quantum number ($m_l$) ranges from $-l$ to $+l$:         $-l \le m_l \le +l$

  • For the first energy level ($n = 1$), the maximum allowed value of $l$ is:         $l = 1 - 1 = 0$

  • An $s$ orbital corresponds to $l = 0$, while a $p$ orbital requires $l = 1$.

  • Because $l = 1$ is forbidden when $n = 1$, a $1p$ orbital cannot exist.

• Mathematical Proof for the Non-Existence of $3f$ Orbitals:

  • An $f$ subshell corresponds to an angular momentum quantum number of $l = 3$

  • For the third energy level ($n = 3$), the maximum allowed value of $l$ is:         $l = 3 - 1 = 2$

  • The allowed subshells for $n = 3$ are $l = 0$ ($s$), $l = 1$ ($p$), and $l = 2$ ($d$).

  • Because $l = 3$ is strictly forbidden when $n = 3$, $3f$ orbitals cannot exist.

• $f$ Orbital Subshell Capacity:

  • $f$ orbitals possess complex three-dimensional spatial geometries.

  • An $f$ subshell contains exactly $7$ degenerate $f$ orbitals ($m_l \in \{-3, -2, -1, 0, 1, 2, 3\}$).

Electron Configurations, Energy Levels, and Ground vs. Excited States

• Electron Configuration Notation:

  • Standard notation lists the principal quantum number ($n$), subshell letter ($s, p, d, f$), and a superscript on the top right denoting the total number of electrons in that subshell.

  • Example (Neon, $Ne$): Neon is a noble gas (inert gas) located in Group $18$ with an atomic number of $10$. Its full electron configuration is:         $1s^2\,2s^2\,2p^6$

• Orbital Energy Filling Order:

  • Electrons fill orbitals in order of increasing energy level, beginning with the lowest energy orbital ($1s$), until all available electrons are assigned.

  • Hydrogen ($H$): Atomic number $1$. Located in the first period in the $s$ subshell. Electron configuration:         $1s^1$

  • Lithium ($Li$): Atomic number $3$. Key component in lithium-ion batteries. Fills the $1s$ subshell before placing its valence electron into the second energy level:         $1s^2\,2s^1$

• Ground State versus Excited State:

  • Ground State: The lowest energy, most stable arrangement of electrons following standard orbital filling principles.

  • Excited State: Occurs when energy or heat stimulates an atom, promoting electrons from lower energy orbitals (e.g., $1s$) to higher energy, unfilled orbitals (e.g., $4f$, $6d$).

Aufbau Principle and Hund's Rule of Maximum Multiplicity

• Limitations of the Aufbau Principle:

  • The Aufbau (building-up) principle specifies the relative energy levels of subshells for electron filling.

  • The Aufbau principle alone does not dictate how electrons are distributed among degenerate (equal energy) orbitals within the same subshell.

• The Problem of Filling Degenerate $p$ Subshells:

  • A $p$ subshell contains $3$ degenerate orbitals ($p_x$, $p_y$, $p_z$).

  • For Carbon ($C$) with $2$ electrons in its $2p$ subshell, two distinct filling arrangements are hypothetically possible:

    1. The two electrons occupy two separate orbitals singly.

    2. The two electrons pair up within a single orbital.

• Hund's Rule of Maximum Multiplicity:

  • Hund's rule dictates that when filling degenerate orbitals, electrons occupy separate orbitals singly with parallel spins before pairing up.

  • This arrangement minimizes electron-electron repulsion and maximizes the total spin quantum number ($S$), making it energy favorable.

• Spin Quantum Calculation Comparison:

  • For paired electrons in the same orbital with opposite spins:         $S = \frac{1}{2} + \left(-\frac{1}{2}\right) = 0$

  • For two electrons in separate orbitals with parallel spins:         $S = \frac{1}{2} + \frac{1}{2} = 1$

  • Because $S = 1$ maximizes spin multiplicity compared to $S = 0$, parallel spin single-occupancy is energy favorable for ground-state configurations.

Electron Filling Applications and Housing Metaphors

• Nitrogen ($N$) Electron Configuration:

  • Nitrogen ($Z = 7$) has $3$ electrons in its $2p$ subshell.

  • Following Hund's rule, each electron occupies a separate $p$ orbital singly with parallel spins:         $1s^2\,2s^2\,2p_x^1\,2p_y^1\,2p_z^1$

• Oxygen ($O$) Electron Configuration:

  • Oxygen ($Z = 8$) has $4$ electrons in its $2p$ subshell.

  • Orbital box diagrams represent each orbital as a box.

  • Because only $3$ degenerate $p$ orbitals exist, the fourth electron must pair up with an electron in one of the occupied orbitals:         $1s^2\,2s^2\,2p^4$

  • Oxygen contains $1$ paired $p$ orbital and $2$ unpaired singly occupied $p$ orbitals.

• Student Housing Metaphor for Electron Repulsion:

  • Electrons behave like university students seeking housing:

    • Students prefer living alone in single-occupancy one-bedroom apartments where they can cook and make noise comfortably without disturbing roommates.

    • Sharing an apartment with a roommate requires sacrificing comfort and increasing energy/tension.

    • When $4$ students must occupy $3$ available rooms, at least two students have no choice but to share a room due to room capacity limits.

Pauli Exclusion Principle and Periodic Table Blocks

• Pauli Exclusion Principle:

  • No two electrons in an atom can have identical sets of all four quantum numbers ($n$, $l$, $m_l$, $m_s$).

  • Each orbital can hold a maximum of $2$ electrons, and they must have opposite (paired) spins ($m_s = +\frac{1}{2}$ and $m_s = -\frac{1}{2}$).

• Post Office Address Metaphor:

  • Delivering mail requires a complete, unique address (State, City, Street, House Number).

  • Similarly, identifying a specific electron in an atom requires a unique combination of four quantum numbers ($n$, $l$, $m_l$, $m_s$).

• Quantum Limits and Subshell Capacities:

  • It is quantum forbidden to place $7$ electrons into a $p$ subshell because its maximum capacity is $6$ electrons ($3$ orbitals \times $2$ electrons per orbital).

  • The $s$-block of the periodic table consists of exactly $2$ columns because an $s$ subshell contains $1$ orbital, which can hold a maximum of $2$ electrons.

• Condensed Noble Gas Notation:

  • Writing full electron configurations for large elements is time-consuming.

  • Condensed notation replaces the inner-shell electron configuration with the bracketed symbol of the preceding noble gas (e.g., $[Ne]$).

Questions and Student Discussion

• Student Course Clarifications and Exam Preparation:

  • Question: Will topics such as noble gas shorthand and electron configurations for larger elements be covered on the upcoming test?

  • Response: The professor confirmed that material covered during the lecture, including electron configurations and orbital rules, will appear on the test.

  • Student Discussion: Students noted that after working through the step-by-step quantum mechanics rules and practice problems, the logic behind orbital filling and periodic blocks becomes clear and manageable.