Atomic Structure and Periodicity Study Guide

Fundamental Quantum Numbers and Wave Mechanics

• Schrödinger Wave Equation Foundation:

  • The spatial behavior and energy states of electrons in an atom are described by the Schrödinger wave equation:     $\hat{H}\psi = E\psi$

  • The spatial wave function $\Psi$ is expressed in spherical polar coordinates as:     $\Psi = \Psi(r, \theta, \varphi)$

  • Solutions to the wave equation yield quantum numbers: a set of integers that completely specify the energy, spatial distribution, shape, and orientation of atomic orbitals.

• Principal Quantum Number ($n$):

  • Takes positive integer values: $n = 1, 2, 3, 4, \dots$

  • Determines the overall size and principal energy level of an orbital.

  • As $n$ increases, the orbital size expands, placing the electron further from the nucleus and increasing its potential energy.

• Angular Momentum Quantum Number ($l$):

  • Takes integral values ranging from $0$ to $n - 1$, yielding $n$ possible values for a shell of level $n$

  • Governs the geometric shape of the atomic orbital (also designated as a subshell).

  • Letters correspond to specific numerical values of $l$:

  • $l = 0 \implies s$

  • $l = 1 \implies p$

  • $l = 2 \implies d$

  • $l = 3 \implies f$

  • $l = 4 \implies g$

  • $l = 5 \implies h$

  • Subshell designations based on principal shell $n$:

  • For $n = 3$: $l = 0, 1, 2$ (corresponding to $3s, 3p, 3d$ subshells).

  • For $n = 4$: $l = 0, 1, 2, 3$ (corresponding to $4s, 4p, 4d, 4f$ subshells).

• Magnetic Quantum Number ($m_l$):

  • Takes integral values from $-l$ to $+l$, including zero.

  • The total number of allowed spatial orientations for a subshell of value $l$ is given by $2l + 1$

  • Specifies the spatial orientation of an orbital relative to the coordinate axes of the atom.

  • Examples:

  • For $l = 1$ ($p$ subshell): $m_l = -1, 0, +1$ (3 possible orientations: $p_x, p_y, p_z$).

  • For $l = 2$ ($d$ subshell): $m_l = -2, -1, 0, +1, +2$ (5 possible orientations, e.g., $4d$ subshell).

Orbital Shapes, Nodal Surfaces, and Energy Levels

• Nodes and Nodal Surfaces:

  • A node or nodal surface is a spatial region where the probability density of finding an electron ($\psi^2$) is exactly zero.

  • The number of nodes increases as the principal quantum number $n$ increases.

  • For spherical $s$ orbitals, the number of radial nodes is calculated as:     $\text{Number of nodes} = n - 1$

  • $1s$ orbital ($n = 1$): $1 - 1 = 0$ nodes.

  • $2s$ orbital ($n = 2$): $2 - 1 = 1$ node.

  • $3s$ orbital ($n = 3$): $3 - 1 = 2$ nodes.

• Characteristics of $s$ Orbitals:

  • Spherical symmetry centered on the atomic nucleus.

  • Electron probability density depends solely on distance $r$ from the nucleus.

• Characteristics of $p$ Orbitals:

  • Non-spherical shape consisting of two distinct lobes situated on opposite sides of a nodal plane passing through the nucleus.

  • Occur in shells where $n \ge 2$ ($l = 1$).

  • The three degenerate $p$ orbitals are aligned mutually perpendicular along Cartesian axes: $p_x, p_y, p_z$

• Characteristics of $d$ Orbitals:

  • First appear in principal quantum level $n = 3$ ($l = 2$, $n_{\text{min}} = 3$); do not exist in $n = 1$ or $n = 2$

  • Five spatial orientations possessing two fundamental shape categories:

  1. Four-lobed structures centered in specific coordinate planes:

     • $d_{xz}$: Lobes centered in the xz-plane between axes.

     • $d_{yz}$: Lobes centered in the yz-plane between axes.

     • $d_{xy}$: Lobes centered in the xy-plane between axes.

     • $d_{x^2 - y^2}$: Lobes centered directly along the x and y axes.

  2. Unique double-lobed doughnut structure:

     • $d_{z^2}$: Two primary lobes pointing along the z-axis encircled by a doughnut-shaped belt centered in the xy plane.

  • For levels where $n > 3$, $d$ orbitals maintain these structural shapes but possess larger lobes.

• Characteristics of $f$ Orbitals:

  • First appear in principal quantum level $n = 4$ ($l = 3$).

  • Seven degenerate orbitals ($m_l = -3, -2, -1, 0, +1, +2, +3$) with complex multi-lobed geometries.

  • Are not involved in chemical bonding in standard chemical compounds.

• Orbital Degeneracy in Single-Electron Systems:

  • Orbitals that share identical energy levels are called degenerate.

  • For single-electron systems such as the hydrogen atom ($\text{H}$), orbital energy $E$ is determined solely by $n$

  • In hydrogen, all orbitals within a given principal shell (such as $3s, 3p, 3d$) are degenerate.

  • In its ground state, the single electron of hydrogen resides in the $1s$ orbital, but it can be excited into higher energy degenerate orbitals.

Electron Spin and the Pauli Exclusion Principle

• Electron Spin Quantum Number ($m_s$):

  • Describes the intrinsic magnetic spin orientation of an electron.

  • Takes only two allowed quantized values:     $m_s = +\frac{1}{2} \quad \text{and} \quad m_s = -\frac{1}{2}$

  • Indicates that electrons spin in two opposite directions, generating opposite magnetic moments.

  • Causes energy level splitting when placed inside an external magnetic field.

• Experimental Proof: The Stern-Gerlach Experiment:

  • Demonstrated electron spin quantization by passing a collimated beam of neutral silver ($\text{Ag}$) atoms through an inhomogeneous magnetic field.

  • The inhomogeneous field splits the neutral atom beam into two distinct trajectories (spin-up and spin-down), verifying the existence of two quantized spin states.

• The Pauli Exclusion Principle:

  • States that in a given atom, no two electrons can have the exact same set of four quantum numbers $(n, l, m_l, m_s)$.

  • Because an individual spatial orbital is defined by $(n, l, m_l)$, an orbital can hold a maximum of two electrons, which must possess opposite spins ($+\frac{1}{2}$ and $-\frac{1}{2}$).

• Summary of Quantum Numbers and Maximum Shell Capacity:

  • $n$ (Principal): Energy and size ($1, 2, 3, \dots$).

  • $l$ (Angular momentum): Shape ($0$ to $n-1$).

  • $m_l$ (Magnetic): Spatial orientation ($-l$ to $+l$).

  • $m_s$ (Spin): Spin direction ($+\frac{1}{2}, -\frac{1}{2}$).

  • Maximum electron capacity for any principal energy level $n$ is given by:     $\text{Maximum capacity} = 2n^2$

  • For $n = 1$: $2(1)^2 = 2$ electrons ($1s$).

  • For $n = 2$: $2(2)^2 = 8$ electrons ($2s, 2p$).

  • For $n = 3$: $2(3)^2 = 18$ electrons ($3s, 3p, 3d$; calculated as 1 $s$-orbital [2 e⁻] + 3 $p$-orbitals [6 e⁻] + 5 $d$-orbitals [10 e⁻]).

Orbital Energies and Penetration in Polyelectronic Atoms

• Polyelectronic Atoms and Electron Correlation:

  • Polyelectronic atoms contain two or more electrons.

  • Involve complex electron-electron repulsions alongside electron-nucleus attractions.

  • Because exact trajectories of electrons are unknown, electron-electron repulsions cannot be precisely calculated (the electron correlation problem).

• Subshell Energy Splitting:

  • Unlike hydrogen, subshell energies within the same principal quantum level $n$ in multi-electron atoms are not degenerate.

  • Orbital levels within a given shell increase in energy in the order:     $E_{ns} < E_{np} < E_{nd} < E_{nf}$

• The Penetration Effect:

  • Describes the ability of an electron to approach close to the nucleus through inner shielding shells.

  • A $2s$ electron penetrates closer to the nucleus than a $2p$ electron due to a small inner probability peak.

  • Greater penetration increases nuclear attraction and lowers potential energy, making $2s$ lower in energy than $2p$

  • In level $n = 3$, penetration capability decreases in the order $3s > 3p > 3d$, giving the relative subshell energy ordering:     $E_{3s} < E_{3p} < E_{3d}$

History of the Periodic Table

• Dmitri Mendeleev's Contributions:

  • Originally constructed the periodic table to reflect recurring patterns in the chemical properties of elements.

  • Used periodic trends to accurately predict the existence and chemical characteristics of undiscovered elements.

  • Corrected previously inaccurate atomic mass values for several known elements.

• Modern Periodic Law:

  • The modern periodic table arranges elements by increasing atomic number ($Z$) rather than atomic mass.

The Aufbau Principle, Hund's Rule, and Electron Configurations

• The Aufbau Principle:

  • Electrons occupy available atomic orbitals in sequence of increasing orbital energy.

• Hund's Rule:

  • States that the lowest energy configuration for an atom is the one with the maximum number of unpaired electrons allowed by the Pauli principle within a degenerate set of orbitals.

  • Unpaired electrons in degenerate orbitals occupy individual orbitals with parallel spins ($m_s$ values of the same sign).

• Electron Configurations and Orbital Diagrams:

  • Nitrogen ($\text{N}$, $Z = 7$):

  • Configuration: $1s^2 2s^2 2p^3$

  • Three unpaired electrons in the $2p$ subshell with parallel spins.     

    

  • Oxygen ($\text{O}$, $Z = 8$):

  • Configuration: $1s^2 2s^2 2p^4$

  • Two electrons in $1s$, two in $2s$, and four in $2p$

  • Neon ($\text{Ne}$, $Z = 10$):

  • Configuration: $1s^2 2s^2 2p^6$

  • Completely filled $n = 1$ and $n = 2$ shells.     

    

• Valence vs. Core Electrons:

  • Valence Electrons: Electrons in the outermost principal quantum level ($n$) of an atom. They dictate chemical reactivity.

  • Core Electrons: Inner shell electrons corresponding to filled noble gas cores.

  • Example - Sodium ($\text{Na}$, $Z = 11$):

  • Configuration: $1s^2 2s^2 2p^6 3s^1$ or $[\text{Ne}]3s^1$

  • Core electrons: $10$ ($1s^2 2s^2 2p^6$

  • Valence electron: $1$ ($3s^1$

• Main Groups and Subshell Blocks:

  • Main Group Elements (Groups 1A–8A / Groups 1, 2, 13–18): Group label number (A designation) equals the total number of valence electrons.

  • Transition Metals: Progressive filling of the five $d$ orbitals (e.g., $3d$).

  • Chromium ($\text{Cr}$, $Z = 24$) configuration: $[\text{Ar}]4s^1 3d^5$

  • Lanthanide Series: Occurs after Lanthanum ($\text{La}$), corresponding to filling seven $4f$ orbitals.

  • Actinide Series: Occurs after Actinium ($\text{Ac}$), corresponding to filling seven $5f$ orbitals.

• Worked Examples of Configurations:

  • Sulfur ($\text{S}$, $Z = 16$):

  • Period 3, Group 6A (16). 4th element in the $3p$ block.

  • Full configuration: $1s^2 2s^2 2p^6 3s^2 3p^4$

  • Shorthand: $[\text{Ne}]3s^2 3p^4$

  • Cadmium ($\text{Cd}$, $Z = 48$):

  • Period 5, end of the $4d$ transition metals (10th $d$-element).

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

  • Shorthand: $[\text{Kr}]5s^2 4d^{10}$

Periodic Trends in Atomic Properties

• Atomic and Ionic Radii Trends:

  • Determination Methods:

  • Covalent atomic radii: Estimated from internuclear distances in covalent molecules.

  • Metallic radii: Half the distance between adjacent nuclei in metallic crystals.

  • Across a Period (Left to Right):

  • Atomic radius decreases.

  • Reason: Effective nuclear charge ($Z_{\text{eff}}$) increases across a period while inner shielding remains constant, pulling valence electrons closer to the nucleus.

  • Down a Group (Top to Bottom):

  • Atomic radius increases.

  • Reason: Valence electrons occupy higher principal quantum levels ($n$) with larger spatial extent.

  • Group 2A Cation Radii Trend:

  • For Group 2A ions ($\text{Be}^{2+}, \text{Mg}^{2+}, \text{Ca}^{2+}, \text{Sr}^{2+}) formed by losing two valence electrons:\n - Radius increases going down the group:\n      \text{Be}^{2+} < \text{Mg}^{2+} < \text{Ca}^{2+} < \text{Sr}^{2+}\n - \text{Be}^{2+}$has the smallest ionic radius;$\text{Sr}^{2+} has the largest.\n\n\n\n- **Ionization Energy (I)**:\n - **Definition**: Energy required to remove an electron from a gaseous atom or ion in its ground state:\n    \text{X}(g) \rightarrow \text{X}^+(g) + e^-\n - **First Ionization Energy (I_1)**: Energy to remove the highest-energy electron from a neutral gaseous atom.\n - **Second Ionization Energy (I_2$)**: Energy to remove an electron from a$1+ cation:\n    \text{X}^+(g) \rightarrow \text{X}^{2+}(g) + e^-\n - I_1 < I_2 because removing an electron from a cation requires overcoming greater electrostatic attraction.\n - **Across a Period (Left to Right)**: First ionization energy generally **increases** due to increasing Z_{\text{eff}}.\n - **Down a Group (Top to Bottom)**: First ionization energy **decreases** because outer electrons are further from the nucleus.\n\n\n\n- **Analysis Case Study on Ionization Energies**:\n - Given configurations:\n 1. 1s^2 2s^2 2p^6$(Neon,$\text{Ne})\n 2. 1s^2 2s^2 2p^6 3s^1$(Sodium,$\text{Na})\n 3. 1s^2 2s^2 2p^6 3s^2$(Magnesium,$\text{Mg})\n - **Largest First Ionization Energy (I_1)**:\n - Neon (1s^2 2s^2 2p^6$) has the largest$I_1\n - *Reason*: Neon is at the right end of Period 2 with a stable, tightly bound filled shell. The 3s electrons of Sodium and Magnesium are shielded by core electrons and lie further from the nucleus.\n - **Smallest Second Ionization Energy (I_2)**:\n - Magnesium (1s^2 2s^2 2p^6 3s^2$) has the smallest$I_2\n - *Reason*: Removing a second electron from Sodium requires removing a core 2p$electron from a stable noble gas configuration ($1s^2 2s^2 2p^6$). For Magnesium, the second electron removed is still a valence$3s$electron ($3s^1 \rightarrow 2p^6$$), requiring substantially less energy.