Comprehensive Study Guide to Periodic Trends: Atomic & Ionic Radii, Ionization Energies, Electron Affinities, and d-Block Cation Configurations

Course Information and Academic Deadlines

• Course Identification & Schedule:

  • Course: Chem 150: Structure & Properties

  • Instructor: Dr. Lili Wang (pronouns: she/her/hers)

  • Lecture Schedule: Tuesday and Thursday (TT) 10:00 AM – 11:15 AM

  • Lecture Location: Atwood Hall, Room 215

• Assignment Deadlines and Course Milestones:

  • Aktiv Chemistry Online Homework: Due on Monday night; accessible via web browsers Google Chrome or Mozilla Firefox.

  • Learning Reflection Survey 1: Hosted on Canvas; due Monday, October 5 at 11:59 PM (remains available until Wednesday, October 7 at 11:59 PM); contributes 10 points ($1\%$ of the total course grade).

  • Exam 1 Regrade Requests: Due in person on Tuesday, October 6 (must be handed directly to the instructor during class).

  • Regrade Request Form Location on Canvas:

  • Canvas Course Path: FA26_CHEM_150_3 > Modules / Shared Files > Chem 150 > Shared to Canvas - Chem 150 > Old Exams and other study materials > Fall 2026.

  • Target Form: RegradeSheet_Fall2026.docx (modified by Tracy McGill).

  • Associated Reference Documents in Folder: Chem150_Exam1_Fall2026_FINAL.pdf.

  • Archived Course Materials on OneDrive/Canvas:

  • Fall 2025 folder (modified September 28, 2025 by Tracy McGill, containing 6 items).

  • Fall 2024 folder (modified February 13, 2025 by James Kindt, containing 11 items).

  • Nomenclature Guide-Chem150.pdf (modified November 5, 2023 by James Kindt, size $130\,\text{KB}$).

  • ECCP - readings that might be helpfu... (modified August 31, 2022 by James Kindt, size $12.8\,\text{KB}$).

  • Quiz 1 (covering Classes 2–5): Due September 18 at 9:00 PM (10 points total).

Fundamental Factors Governing Orbital Size and Atomic Radii

• Physical Determinants of Orbital Size:

  • Principal Quantum Number ($n$) and Subshell Type ($l$): Orbital size is predominantly governed by the principal quantum shell $n$. Electrons residing in higher $n$ levels possess greater average radial distance from the nucleus. Subshell identity ($s$, $p$, or $d$) also introduces minor variations due to differing angular wavefunctions and radial penetration abilities.

  • Nuclear Charge ($Z$): The total positive charge produced by protons in the nucleus. A higher nuclear charge exerts a stronger inward electrostatic attraction on the electron distribution.

  • Screening (Shielding) and Electron-Electron Repulsion ($S$): Inner-shell (core) electrons physically shield valence electrons from experiencing the full electrostatic force of the nuclear charge, while mutual electrostatic repulsion among valence electrons pushes them farther outward.

  • Effective Nuclear Charge ($Z_{\text{eff}}$): Combines the opposing effects of nuclear charge $Z$ and electron screening $S$:     $Z_{\text{eff}} = Z - S$

• Periodic Trends in Atomic Radii:

  • Down a Group (Top to Bottom):

  • Trend: Atomic radius increases progressively.

  • Mechanism: The principal quantum number $n$ increases with each added row. Although nuclear charge $Z$ rises simultaneously, the addition of completely filled inner electron shells markedly increases shielding ($S$), ensuring that valence electrons reside in higher-energy orbitals situated farther from the nucleus.

  • Across a Period (Left to Right):

  • Trend: Atomic radius decreases steadily.

  • Mechanism: Electrons are added sequentially to the same principal shell ($n$ is constant), where they shield one another very poorly. Concurrently, protons are added to the nucleus one by one, raising nuclear charge $Z$. Because $Z$ increases faster than shielding $S$, effective nuclear charge $Z_{\text{eff}}$ rises, pulling the electron cloud inward.

• Quantitative Atomic Radii Data (in picometers, $\text{pm}$):

  • Period 1:

  • $\text{H}$ ($37\,\text{pm}$

  • $\text{He}$ ($31\,\text{pm}$

  • Period 2:

  • $\text{Li}$ ($152\,\text{pm}$

  • $\text{Be}$ ($112\,\text{pm}$

  • $\text{B}$ ($85\,\text{pm}$

  • $\text{C}$ ($77\,\text{pm}$

  • $\text{N}$ ($75\,\text{pm}$

  • $\text{O}$ ($73\,\text{pm}$

  • $\text{F}$ ($72\,\text{pm}$

  • $\text{Ne}$ ($71\,\text{pm}$

  • Period 3:

  • $\text{Na}$ ($186\,\text{pm}$

  • $\text{Mg}$ ($160\,\text{pm}$

  • $\text{Al}$ ($143\,\text{pm}$

  • $\text{Si}$ ($118\,\text{pm}$

  • $\text{P}$ ($110\,\text{pm}$

  • $\text{S}$ ($103\,\text{pm}$

  • $\text{Cl}$ ($100\,\text{pm}$

  • $\text{Ar}$ ($98\,\text{pm}$

  • Period 4 (Main Group):

  • $\text{K}$ ($227\,\text{pm}$

  • $\text{Ca}$ ($197\,\text{pm}$

  • $\text{Ga}$ ($135\,\text{pm}$

  • $\text{Ge}$ ($122\,\text{pm}$

  • $\text{As}$ ($120\,\text{pm}$

  • $\text{Se}$ ($119\,\text{pm}$

  • $\text{Br}$ ($114\,\text{pm}$

  • $\text{Kr}$ ($112\,\text{pm}$

  • Period 5 (Main Group):

  • $\text{Rb}$ ($248\,\text{pm}$

  • $\text{Sr}$ ($215\,\text{pm}$

  • $\text{In}$ ($167\,\text{pm}$

  • $\text{Sn}$ ($140\,\text{pm}$

  • $\text{Sb}$ ($140\,\text{pm}$

  • $\text{Te}$ ($142\,\text{pm}$

  • $\text{I}$ ($133\,\text{pm}$

  • $\text{Xe}$ ($131\,\text{pm}$

  • Period 6 (Main Group):

  • $\text{Cs}$ ($265\,\text{pm}$

  • $\text{Ba}$ ($222\,\text{pm}$

  • $\text{Tl}$ ($170\,\text{pm}$

  • $\text{Pb}$ ($146\,\text{pm}$

  • $\text{Bi}$ ($150\,\text{pm}$

  • $\text{Po}$ ($168\,\text{pm}$

  • $\text{At}$ ($140\,\text{pm}$

  • $\text{Rn}$ ($140\,\text{pm}$

  • Period 7:

  • $\text{Fr}$ ($270\,\text{pm}$

  • $\text{Ra}$ ($220\,\text{pm}$

• Analysis of the Radius Drop Between Lithium and Beryllium:

  • Lithium ($\text{Li}$, $Z = +3$): Contains configuration $1s^2 2s^1$. The single $2s$ valence electron experiences substantial shielding by the two inner $1s$ core electrons. Thus, the effective nuclear charge felt by the outermost electron is close to $+1$, yielding an expanded radius of $152\,\text{pm}$.

  • Beryllium ($\text{Be}$, $Z = +4$): Contains configuration $1s^2 2s^2$. Although shielded by two $1s$ electrons, the second $2s$ electron shields the other $2s$ electron very poorly. The addition of a proton to the nucleus increases inward electrostatic attraction far more than inter-electron repulsion increases. As a result, $Z_{\text{eff}}$ rises substantially, contracting the atomic radius to $112\,\text{pm}$ (a net reduction of $40\,\text{pm}$).

Ionic Radii Trends and Isoelectronic Comparisons

• Principles of Cation Formation and Size Reduction:

  • Loss of Valence Electrons: Removing valence electrons reduces mutual electron-electron repulsions, allowing the remaining electrons to be pulled closer to the positively charged nucleus.

  • Complete Shell Vacatur: When all valence electrons are lost, the entire outermost principal shell ($n$) is emptied, leaving the smaller, tightly held core shell as the outermost layer.

  • Quantitative Neutral Atom vs. Cation Radii ($\text{pm}$):

  • Group 1A:

    • $\text{Li}$ ($152\,\text{pm}$) $\rightarrow$ $\text{Li}^+$ ($60\,\text{pm}$

    • $\text{Na}$ ($186\,\text{pm}$) $\rightarrow$ $\text{Na}^+$ ($95\,\text{pm}$

    • $\text{K}$ ($227\,\text{pm}$) $\rightarrow$ $\text{K}^+$ ($133\,\text{pm}$

    • $\text{Rb}$ ($248\,\text{pm}$) $\rightarrow$ $\text{Rb}^+$ ($148\,\text{pm}$

  • Group 2A:

    • $\text{Be}$ ($112\,\text{pm}$) $\rightarrow$ $\text{Be}^{2+}$ ($31\,\text{pm}$

    • $\text{Mg}$ ($160\,\text{pm}$) $\rightarrow$ $\text{Mg}^{2+}$ ($65\,\text{pm}$

    • $\text{Ca}$ ($197\,\text{pm}$) $\rightarrow$ $\text{Ca}^{2+}$ ($99\,\text{pm}$

    • $\text{Sr}$ ($215\,\text{pm}$) $\rightarrow$ $\text{Sr}^{2+}$ ($113\,\text{pm}$

  • Group 3A:

    • $\text{B}$ ($85\,\text{pm}$) $\rightarrow$ $\text{B}^{3+}$ ($23\,\text{pm}$

    • $\text{Al}$ ($143\,\text{pm}$) $\rightarrow$ $\text{Al}^{3+}$ ($50\,\text{pm}$

    • $\text{Ga}$ ($135\,\text{pm}$) $\rightarrow$ $\text{Ga}^{3+}$ ($62\,\text{pm}$

    • $\text{In}$ ($166\,\text{pm}$) $\rightarrow$ $\text{In}^{3+}$ ($81\,\text{pm}$

• Principles of Anion Formation and Size Expansion:

  • Addition of Valence Electrons: Gaining electrons into the valence shell increases the total magnitude of electron-electron repulsion without altering the nuclear charge $Z$.

  • The electron cloud expands outward to minimize Coulombic repulsion between adjacent electrons.

  • Quantitative Neutral Atom vs. Anion Radii ($\text{pm}$):

  • Group 6A:

    • $\text{O}$ ($73\,\text{pm}$) $\rightarrow$ $\text{O}^{2-}$ ($140\,\text{pm}$

    • $\text{S}$ ($103\,\text{pm}$) $\rightarrow$ $\text{S}^{2-}$ ($184\,\text{pm}$

    • $\text{Se}$ ($117\,\text{pm}$) $\rightarrow$ $\text{Se}^{2-}$ ($198\,\text{pm}$

    • $\text{Te}$ ($143\,\text{pm}$) $\rightarrow$ $\text{Te}^{2-}$ ($221\,\text{pm}$

  • Group 7A:

    • $\text{F}$ ($72\,\text{pm}$) $\rightarrow$ $\text{F}^-$ ($136\,\text{pm}$

    • $\text{Cl}$ ($99\,\text{pm}$) $\rightarrow$ $\text{Cl}^-$ ($181\,\text{pm}$

    • $\text{Br}$ ($114\,\text{pm}$) $\rightarrow$ $\text{Br}^-$ ($195\,\text{pm}$

    • $\text{I}$ ($133\,\text{pm}$) $\rightarrow$ $\text{I}^-$ ($216\,\text{pm}$

• Worksheet 9 Part IV: Size Ranking Problems and Scientific Explanations:

  • Set a: Ge, Si, Sn

  • Decreasing Order: $\text{Sn} > \text{Ge} > \text{Si}$

  • Reasoning: All three elements belong to Group 4A (14). Descending the column increases the principal quantum number $n$ ($n=3$ for $\text{Si}$, $n=4$ for $\text{Ge}$, $n=5$ for $\text{Sn}$), introducing more filled core shells that increase shielding and place valence electrons at larger radial distances.

  • Set b: Sn, Te, Sr

  • Decreasing Order: $\text{Sr} > \text{Sn} > \text{Te}$

  • Reasoning: All three elements reside in Period 5 ($n=5$ valence shell). Moving across the period from left to right, nuclear charge increases ($Z=38$ for $\text{Sr}$, $Z=50$ for $\text{Sn}$, $Z=52$ for $\text{Te}$). Because the added electrons occupy the same shell, they provide minimal shielding against the higher nuclear charge, leading to elevated $Z_{\text{eff}}$ that contracts the radius.

  • Set c: $\text{Se}^{2-}$, $\text{S}^{2-}$, $\text{O}^{2-}$

  • Decreasing Order: $\text{Se}^{2-} > \text{S}^{2-} > \text{O}^{2-}$

  • Reasoning: These ions are Group 6A dianions with identical valence configurations ($s^2 p^6$). As one descends Group 6A, the valence electrons occupy larger principal quantum shells ($n=2$ for $\text{O}^{2-}$, $n=3$ for $\text{S}^{2-}$, $n=4$ for $\text{Se}^{2-}$), systematically increasing ionic radius.

  • Set d: $\text{Te}^{2-}$, $\text{Cs}^+$, $\text{I}^-$

  • Decreasing Order: $\text{Te}^{2-} > \text{I}^- > \text{Cs}^+$

  • Reasoning (Isoelectronic Series): Each of these chemical species possesses exactly 54 electrons, matching the ground-state electron configuration of Xenon ($[\text{Xe}]$). Since inter-electron repulsion and shielding are approximately identical across the series, the size is determined entirely by nuclear charge $Z$:

    • $\text{Te}^{2-}$: $Z = 52$

    • $\text{I}^-$: $Z = 53$

    • $\text{Cs}^+$: $Z = 55$

  • Cesium possesses the highest nuclear charge ($Z=55$), pulling the 54 electrons most tightly inward, whereas Tellurium has the lowest nuclear charge ($Z=52$), resulting in the weakest electrostatic hold and the largest radius.

Lanthanide Contraction and Transition Metal Radii Anomalies

• Worksheet 9 Part III: Group Comparison Across Rows 4, 5, and 6:

  • Standard Expectation: Atomic radii should increase monotonically descending a group due to the increment of principal quantum number $n$.

  • Experimental Radii in Group 3B (3) versus Group 4B (4):

  • Group 3B: $\text{Sc}$ ($157\,\text{pm}$) $\rightarrow$ $\text{Y}$ ($169.3\,\text{pm}$) $\rightarrow$ $\text{La}$ ($191.5\,\text{pm}$) [follows expected increase]

  • Group 4B: $\text{Ti}$ ($147.7\,\text{pm}$) $\rightarrow$ $\text{Zr}$ ($159.3\,\text{pm}$) $\rightarrow$ $\text{Hf}$ ($147.6\,\text{pm}$) [breaks the rule: $\text{Hf} < \text{Zr}$]

• Lanthanide Contraction Phenomenon:

  • Claim: Outermost valence electrons in Hafnium ($\text{Hf}$) experience a significantly higher effective nuclear charge ($Z_{\text{eff}}$) than outermost electrons in Lanthanum ($\text{La}$) or Zirconium ($\text{Zr}$.

  • Evidence: The atomic radius of Hafnium ($147.6\,\text{pm}$) is smaller than that of Zirconium ($159.3\,\text{pm}$), even though Hafnium has an additional principal quantum shell ($n=6$ versus $n=5$).

  • Reasoning:

  • Between Lanthanum ($Z=57$) and Hafnium ($Z=72$), the 14 elements of the lanthanide series fill the interior $4f$ subshell.

  • The $f$ orbitals are diffuse, with node topologies that do not penetrate close to the nucleus. As a consequence, $4f$ electrons are ineffective at shielding outer valence electrons from nuclear charge.

  • Over these 14 elements, the nuclear charge increases by $+14$ without a commensurate increase in shielding. Therefore, valence electrons experience a sharp increase in $Z_{\text{eff}}$, pulling the outer orbitals inward and compensating for the addition of the $n=6$ shell.

Ionization Energy: Fundamentals and Electrostatic Principles

• Verbatim Definition of Ionization Energy (IE):

  • The ionization energy of an atom or ion is the least amount of energy required to completely remove an electron from 1 mole of that gaseous species, when it is in its ground state.

• Chemical Representation and Endothermic Nature:   $\text{Na}(g) \rightarrow \text{Na}^+(g) + e^-\quad \text{IE}_1 = 495.8\,\text{kJ/mol}$

  • The reaction produces a chemical cation with a net positive charge.

  • Ionization energies for neutral atoms are universally positive (endothermic), because electrostatic work must be done to pull a negatively charged electron away from the attractive Coulombic potential of a positively charged nucleus.

• Application of Coulomb's Law:

  • The electrostatic force $F$ between the nucleus ($q_1 = +Z_{\text{eff}}e$) and an electron ($q_2 = -e$) separated by radial distance $r$ is defined by:     $F = k \frac{|q_1 q_2|}{r^2}$

  • As $Z_{\text{eff}}$ increases, attraction increases, raising the ionization energy.

  • As distance $r$ increases (higher $n$), attraction decreases according to the inverse-square relationship, lowering the ionization energy.

• Helium Case Study in Shielding and Ionization Energy:

  • Neutral Helium ($\text{He}$, $Z = +2$) contains two ground-state $1s$ electrons.

  • First Ionization Energy ($\text{IE}_1$): Removing the first electron requires $3.94 \times 10^{-18}\,\text{J}$ per atom ($2372\,\text{kJ/mol}$). Mutual shielding between the two $1s$ electrons lowers the effective nuclear charge felt by either electron below the nominal $+2$.

  • Second Ionization Energy ($\text{IE}_2$): Removing the second electron from $\text{He}^+$ requires $8.72 \times 10^{-18}\,\text{J}$ per atom ($5250\,\text{kJ/mol}$).

  • Explanation: Once the first electron is ejected, all electron-electron repulsion and shielding vanish. The lone remaining electron experiences the unattenuated $+2$ nuclear charge at a contracted orbital radius, more than doubling the energy necessary for ionization.

Periodic Trends in First Ionization Energy ($\text{IE}_1$)

• General Trends Across and Down the Periodic Table:

  • Across a Period (Left to Right): $\text{IE}_1$ generally increases. Nuclear charge $Z$ increases while core shielding remains roughly equivalent, elevating $Z_{\text{eff}}$. Atomic radius decreases, binding valence electrons more tightly to the nucleus.

  • Down a Group (Top to Bottom): $\text{IE}_1$ generally decreases. Principal quantum number $n$ increases, positioning valence electrons farther from the nucleus. Increased core-shell shielding reduces electrostatic hold, making electron removal easier.

• Experimental $\text{IE}_1$ Values for Main Group Elements ($\text{kJ/mol}$):

  • Period 1:

  • $\text{H}$: $1312\,\text{kJ/mol}$

  • $\text{He}$: $2372\,\text{kJ/mol}$

  • Period 2:

  • $\text{Li}$: $520\,\text{kJ/mol}$

  • $\text{Be}$: $899\,\text{kJ/mol}$

  • $\text{B}$: $800\,\text{kJ/mol}$

  • $\text{C}$: $1086\,\text{kJ/mol}$

  • $\text{N}$: $1402\,\text{kJ/mol}$

  • $\text{O}$: $1314\,\text{kJ/mol}$

  • $\text{F}$: $1681\,\text{kJ/mol}$

  • $\text{Ne}$: $2080\,\text{kJ/mol}$

  • Period 3:

  • $\text{Na}$: $496\,\text{kJ/mol}$

  • $\text{Mg}$: $738\,\text{kJ/mol}$

  • $\text{Al}$: $577\,\text{kJ/mol}$

  • $\text{Si}$: $786\,\text{kJ/mol}$

  • $\text{P}$: $1012\,\text{kJ/mol}$

  • $\text{S}$: $999\,\text{kJ/mol}$

  • $\text{Cl}$: $1256\,\text{kJ/mol}$

  • $\text{Ar}$: $1520\,\text{kJ/mol}$

  • Period 4:

  • $\text{K}$: $419\,\text{kJ/mol}$

  • $\text{Ca}$: $590\,\text{kJ/mol}$

  • $\text{Ga}$: $579\,\text{kJ/mol}$

  • $\text{Ge}$: $761\,\text{kJ/mol}$

  • $\text{As}$: $947\,\text{kJ/mol}$

  • $\text{Se}$: $941\,\text{kJ/mol}$

  • $\text{Br}$: $1143\,\text{kJ/mol}$

  • $\text{Kr}$: $1351\,\text{kJ/mol}$

  • Period 5:

  • $\text{Rb}$: $403\,\text{kJ/mol}$

  • $\text{Sr}$: $549\,\text{kJ/mol}$

  • $\text{In}$: $558\,\text{kJ/mol}$

  • $\text{Sn}$: $708\,\text{kJ/mol}$

  • $\text{Sb}$: $834\,\text{kJ/mol}$

  • $\text{Te}$: $869\,\text{kJ/mol}$

  • $\text{I}$: $1009\,\text{kJ/mol}$

  • $\text{Xe}$: $1170\,\text{kJ/mol}$

  • Period 6:

  • $\text{Cs}$: $376\,\text{kJ/mol}$

  • $\text{Ba}$: $503\,\text{kJ/mol}$

  • $\text{Tl}$: $589\,\text{kJ/mol}$

  • $\text{Pb}$: $715\,\text{kJ/mol}$

  • $\text{Bi}$: $703\,\text{kJ/mol}$

  • $\text{Po}$: $813\,\text{kJ/mol}$

  • $\text{At}$: $926\,\text{kJ/mol}$

  • $\text{Rn}$: $1037\,\text{kJ/mol}$

Discontinuities and Exceptions in First Ionization Energy Trends

• Exception 1: Group 2A to Group 3A (Beryllium vs. Boron, Magnesium vs. Aluminum):

  • Observed Discontinuity:

  • $\text{IE}_1(\text{Be}) = 899\,\text{kJ/mol}$ is greater than $\text{IE}_1(\text{B}) = 800\,\text{kJ/mol}$.

  • $\text{IE}_1(\text{Mg}) = 738\,\text{kJ/mol}$ is greater than $\text{IE}_1(\text{Al}) = 577\,\text{kJ/mol}$.

  • Orbital Configurations:

  • $\text{Be}$: $1s^2 2s^2$

  • $\text{B}$: $1s^2 2s^2 2p^1$

  • Explanation via Radial Probability and Penetration:

  • In a given principal quantum shell, orbitals with higher angular momentum quantum numbers ($l$) reside at higher energy states: $E_{ns} < E_{np} < E_{nd} < E_{nf}$.

  • A $2s$ orbital ($l=0$) exhibits penetration close to the nucleus, displaying an inner radial probability maximum near $r/a \approx 0.8$.

  • A $2p$ orbital ($l=1$) possesses an angular node at the nucleus ($r=0$) and cannot penetrate the core as effectively. As a result, the lone $2p$ electron of Boron is partially shielded by the filled inner $2s^2$ subshell in addition to the $1s^2$ core. Being higher in energy and more shielded, this outer $2p$ electron is easier to eject than a paired $2s$ electron from Beryllium.

• Exception 2: Group 5A to Group 6A (Nitrogen vs. Oxygen, Phosphorus vs. Sulfur):

  • Observed Discontinuity:

  • $\text{IE}_1(\text{N}) = 1402\,\text{kJ/mol}$ is greater than $\text{IE}_1(\text{O}) = 1314\,\text{kJ/mol}$.

  • $\text{IE}_1(\text{P}) = 1012\,\text{kJ/mol}$ is greater than $\text{IE}_1(\text{S}) = 999\,\text{kJ/mol}$.

  • Orbital Configurations:

  • $\text{N}$: $[\text{He}]\,2s^2 2p_x^1 2p_y^1 2p_z^1$ (half-filled subshell with parallel spins by Hund's rule)

  • $\text{O}$: $[\text{He}]\,2s^2 2p_x^2 2p_y^1 2p_z^1$ (one doubly occupied orbital)

  • Explanation via Electron-Electron Pairing Repulsion:

  • In Nitrogen, each $2p$ orbital is singly occupied. Electrons maximize spatial separation, minimizing repulsion.

  • In Oxygen, the fourth $2p$ electron must pair with an existing electron in the same orbital.

  • Confining two negatively charged electrons into the same spatial orbital generates localized electrostatic repulsion (pairing energy). This repulsion destabilizes the paired electron, requiring less external energy to remove than an electron from the half-filled, uncoupled $p$ subshell of Nitrogen.

Successive Ionization Energies and Core vs. Valence Discontinuities

• Principles Governing Successive Ionization Energies:

  • Successive ionization energies always increase monotonically:     $\text{IE}_1 < \text{IE}_2 < \text{IE}_3 < \dots < \text{IE}_n$

  • Each successive ionization removes an electron from a cation with a progressively higher positive net charge.

  • Removing an electron leaves behind fewer electrons, diminishing mutual shielding and contracting the remaining electron cloud closer to the nucleus.

• Sodium ($\text{Na}$) versus Magnesium ($\text{Mg}$) Successive Ionization Energies:

  • Sodium Data ($Z=11$, configuration $[\text{Ne}]\,3s^1$):

  • $\text{IE}_1 = 495.8\,\text{kJ/mol}$

  • $\text{IE}_2 = 4562\,\text{kJ/mol}$

  • $\text{IE}_3 = 6910.3\,\text{kJ/mol}$

  • Analysis: The first ionization removes the single $3s$ valence electron. The second ionization breaks into the noble-gas $2p^6$ core shell, resulting in an energy jump of $4066.2\,\text{kJ/mol}$ (a $9.2\times$ increase).

  • Magnesium Data ($Z=12$, configuration $[\text{Ne}]\,3s^2$):

  • $\text{IE}_1 = 737.7\,\text{kJ/mol}$

  • $\text{IE}_2 = 1450.7\,\text{kJ/mol}$

  • $\text{IE}_3 = 7732.7\,\text{kJ/mol}$

  • Analysis: The first and second ionizations remove valence electrons from the $3s$ subshell, with a moderate increase between $\text{IE}_1$ and $\text{IE}_2$. The third ionization requires breaking into the $2p^6$ core shell, triggering an energy jump of $6282.0\,\text{kJ/mol}$ (a $5.3\times$ increase).

• Comprehensive Successive Ionization Energies (Table 4.2, Values in $\text{kJ/mol}$):

  • $\text{Li}$ ($Z=3$):

  • $\text{IE}_1 = 520$

  • $\text{IE}_2 = 7298$

  • $\text{IE}_3 = 11815$

  • *Core electron threshold: $\text{IE}_2$

  • $\text{Be}$ ($Z=4$):

  • $\text{IE}_1 = 899$

  • $\text{IE}_2 = 1757$

  • $\text{IE}_3 = 14848$

  • $\text{IE}_4 = 21007$

  • *Core electron threshold: $\text{IE}_3$

  • $\text{B}$ ($Z=5$):

  • $\text{IE}_1 = 800$

  • $\text{IE}_2 = 2427$

  • $\text{IE}_3 = 3660$

  • $\text{IE}_4 = 25026$

  • $\text{IE}_5 = 32827$

  • *Core electron threshold: $\text{IE}_4$

  • $\text{C}$ ($Z=6$):

  • $\text{IE}_1 = 1086$

  • $\text{IE}_2 = 2353$

  • $\text{IE}_3 = 4621$

  • $\text{IE}_4 = 6223$

  • $\text{IE}_5 = 37831$

  • $\text{IE}_6 = 47277$

  • *Core electron threshold: $\text{IE}_5$

  • $\text{N}$ ($Z=7$):

  • $\text{IE}_1 = 1402$

  • $\text{IE}_2 = 2856$

  • $\text{IE}_3 = 4578$

  • $\text{IE}_4 = 7475$

  • $\text{IE}_5 = 9445$

  • $\text{IE}_6 = 53267$

  • $\text{IE}_7 = 64360$

  • *Core electron threshold: $\text{IE}_6$

  • $\text{O}$ ($Z=8$):

  • $\text{IE}_1 = 1314$

  • $\text{IE}_2 = 3388$

  • $\text{IE}_3 = 5301$

  • $\text{IE}_4 = 7469$

  • $\text{IE}_5 = 10990$

  • $\text{IE}_6 = 13327$

  • $\text{IE}_7 = 71330$

  • $\text{IE}_8 = 84078$

  • *Core electron threshold: $\text{IE}_7$

  • $\text{F}$ ($Z=9$):

  • $\text{IE}_1 = 1681$

  • $\text{IE}_2 = 3374$

  • $\text{IE}_3 = 6050$

  • $\text{IE}_4 = 8408$

  • $\text{IE}_5 = 11023$

  • $\text{IE}_6 = 15164$

  • $\text{IE}_7 = 17868$

  • $\text{IE}_8 = 92038$

  • $\text{IE}_9 = 106434$

  • *Core electron threshold: $\text{IE}_8$

  • $\text{Ne}$ ($Z=10$):

  • $\text{IE}_1 = 2080$

  • $\text{IE}_2 = 3952$

  • $\text{IE}_3 = 6122$

  • $\text{IE}_4 = 9371$

  • $\text{IE}_5 = 12177$

  • $\text{IE}_6 = 15238$

  • $\text{IE}_7 = 19999$

  • $\text{IE}_8 = 23069$

  • $\text{IE}_9 = 115380$

  • $\text{IE}_{10} = 131432$

  • *Core electron threshold: $\text{IE}_9$

  • $\text{Na}$ ($Z=11$):

  • $\text{IE}_1 = 496$

  • $\text{IE}_2 = 4562$

  • $\text{IE}_3 = 6910$

  • $\text{IE}_4 = 9543$

  • $\text{IE}_5 = 13354$

  • $\text{IE}_6 = 16613$

  • $\text{IE}_7 = 20117$

  • $\text{IE}_8 = 25496$

  • $\text{IE}_9 = 28932$

  • $\text{IE}_{10} = 141362$

  • *Core electron threshold: $\text{IE}_2$

• Mechanistic Basis for Core Ionization Jumps:

  • Core electrons reside in a shell with a lower principal quantum number ($n$), positioning them closer to the nucleus.

  • Core electrons encounter fewer filled shells shielding them from the nucleus, resulting in an elevated effective nuclear charge ($Z_{\text{eff}}$).

Electron Affinity Principles and Periodic Behavior

• Definition and Sign Conventions:

  • Definition: Electron affinity (EA) is defined as the energy change accompanying the addition of an electron to a gaseous atom or ion in its ground state:     $\text{X}(g) + e^- \rightarrow \text{X}^-(g)$

  • Thermodynamic Behavior:

  • Exothermic (Energy Released): $\text{EA} < 0$ (conventionally reported in tabulated energy-release charts with positive values, e.g., $\text{Cl} = +349\,\text{kJ/mol}$).

  • Endothermic (Energy Required): Energy must be forced into the system to bind an electron (reported with negative values, e.g., noble gases).

  • Zero / Energetically Neutr al: The addition process incurs zero net energy change.

• Periodic Trends for Electron Affinity:

  • Across a Period (Left to Right): Electron affinity generally becomes more favorable (more energy released) as $Z_{\text{eff}}$ increases, allowing added electrons to bind more tightly.

  • Subshell Discontinuities:

  • Group 2A ($ns^2$): Filled $s$ subshells force an incoming electron into a higher-energy $p$ orbital, making addition unfavorable ($\text{Be} \le 0$, $\text{Mg} \le 0$).

  • Group 5A ($ns^2 np^3$): Half-filled $p$ subshells force an incoming electron to pair with an existing electron, incurring inter-electron repulsion ($\text{N} = -7\,\text{kJ/mol}$).

  • Group 8A ($ns^2 np^6$): Completely filled valence shells force the electron into an entirely new shell ($n+1$), making addition strongly endothermic ($\text{Ne} = -29\,\text{kJ/mol}$, $\text{Ar} = -35\,\text{kJ/mol}$).

• Experimental Electron Affinity Values ($\text{kJ/mol}$ released):

  • Group 1A:

  • $\text{H}$: $+72.8$

  • $\text{Li}$: $+59.6$

  • $\text{Na}$: $+52.9$

  • $\text{K}$: $+48.4$

  • $\text{Rb}$: $+46.9$

  • $\text{Cs}$: $+45.5$

  • Group 2A:

  • $\text{Be}$: $\le 0$

  • $\text{Mg}$: $\le 0$

  • $\text{Ca}$: $+2.37$

  • $\text{Sr}$: $+5.03$

  • $\text{Ba}$: $+13.95$

  • Group 3A:

  • $\text{B}$: $+26.7$

  • $\text{Al}$: $+42.5$

  • $\text{Ga}$: $+28.9$

  • $\text{In}$: $+28.9$

  • $\text{Tl}$: $+19.3$

  • Group 4A:

  • $\text{C}$: $+122$

  • $\text{Si}$: $+134$

  • $\text{Ge}$: $+119$

  • $\text{Sn}$: $+107$

  • $\text{Pb}$: $+35.1$

  • Group 5A:

  • $\text{N}$: $-7$

  • $\text{P}$: $+72.0$

  • $\text{As}$: $+78.2$

  • $\text{Sb}$: $+103$

  • $\text{Bi}$: $+91.3$

  • Group 6A:

  • $\text{O}$: $+141$

  • $\text{S}$: $+200$

  • $\text{Se}$: $+195$

  • $\text{Te}$: $+190$

  • $\text{Po}$: $+183$

  • Group 7A (Halogens):

  • $\text{F}$: $+328$

  • $\text{Cl}$: $+349$

  • $\text{Br}$: $+325$

  • $\text{I}$: $+295$

  • $\text{At}$: $+270$

  • Group 8A (Noble Gases):

  • $\text{He}$: $0.0$

  • $\text{Ne}$: $-29$

  • $\text{Ar}$: $-35$

  • $\text{Kr}$: $-39$

  • $\text{Xe}$: $-41$

  • $\text{Rn}$: $-41$

Electronic Structure and Ionization of Transition Metals ($d$-Block)

• Orbital Energy Inversion Across Period 4:

  • Hydrogen Atom Baseline: Orbitals within a principal shell are degenerate:     $4d = 4p = 4s > 3d = 3p = 3s$

  • Neutral Multi-electron Atoms: Subshell energy follows $3s < 3p < 3d$. At Potassium ($Z=19$) and Calcium ($Z=20$), the $4s$ orbital penetrates near the nucleus, making it slightly lower in energy than the unshielded $3d$ orbital ($E_{4s} < E_{3d}$). Thus, $4s$ fills first.

  • Orbital Energy Crossing Point: At the onset of the transition series ($Z \ge 21$, Scandium), the added nuclear charge causes $3d$ orbitals to experience a rapid drop in potential energy. Consequently, the $3d$ subshell falls below the $4s$ orbital in energy ($E_{3d} < E_{4s}$).

• Complete Ground-State Electron Configurations for Transition Metals:

  • Period 4 ($3d$ Series):

  • $\text{Sc}$ ($Z=21$): $[\text{Ar}]\,4s^2 3d^1$

  • $\text{Ti}$ ($Z=22$): $[\text{Ar}]\,4s^2 3d^2$

  • $\text{V}$ ($Z=23$): $[\text{Ar}]\,4s^2 3d^3$

  • $\text{Cr}$ ($Z=24$): $[\text{Ar}]\,4s^1 3d^5$ (half-filled stability exception)

  • $\text{Mn}$ ($Z=25$): $[\text{Ar}]\,4s^2 3d^5$

  • $\text{Fe}$ ($Z=26$): $[\text{Ar}]\,4s^2 3d^6$

  • $\text{Co}$ ($Z=27$): $[\text{Ar}]\,4s^2 3d^7$

  • $\text{Ni}$ ($Z=28$): $[\text{Ar}]\,4s^2 3d^8$

  • $\text{Cu}$ ($Z=29$): $[\text{Ar}]\,4s^1 3d^{10}$ (filled $d$-subshell stability exception)

  • $\text{Zn}$ ($Z=30$): $[\text{Ar}]\,4s^2 3d^{10}$

  • Period 5 ($4d$ Series):

  • $\text{Y}$ ($Z=39$): $[\text{Kr}]\,5s^2 4d^1$

  • $\text{Zr}$ ($Z=40$): $[\text{Kr}]\,5s^2 4d^2$

  • $\text{Nb}$ ($Z=41$): $[\text{Kr}]\,5s^1 4d^4$

  • $\text{Mo}$ ($Z=42$): $[\text{Kr}]\,5s^1 4d^5$

  • $\text{Tc}$ ($Z=43$): $[\text{Kr}]\,5s^2 4d^5$

  • $\text{Ru}$ ($Z=44$): $[\text{Kr}]\,5s^1 4d^7$

  • $\text{Rh}$ ($Z=45$): $[\text{Kr}]\,5s^1 4d^8$

  • $\text{Pd}$ ($Z=46$): $[\text{Kr}]\,4d^{10}$ (zero $5s$ electrons)

  • $\text{Ag}$ ($Z=47$): $[\text{Kr}]\,5s^1 4d^{10}$

  • $\text{Cd}$ ($Z=48$): $[\text{Kr}]\,5s^2 4d^{10}$

• Ionization Mechanism for Transition Metal Cations:

  • The Highest-$n$ Rule: When forming transition metal cations, electrons are removed first from the subshell with the highest principal quantum number ($n$).

  • For Period 4 transition metals, valence electrons are ionized from the $4s$ subshell before any electrons are removed from the $3d$ subshell.

  • Physical Cause: In a cation, net positive charge increases, causing all orbitals to contract. The compact $3d$ orbitals contract closer to the nucleus than the diffuse $4s$ orbital, experiencing a large stabilizing energy drop ($E_{3d} \ll E_{4s}$). Consequently, the remaining $4s$ electrons are held at higher energy and are lost first.

  • Iron Ionization Pathway:

  • Neutral Iron: $\text{Fe}\quad [\text{Ar}]\,4s^2 3d^6$

  • Iron(II) Cation: $\text{Fe}^{2+}\quad [\text{Ar}]\,3d^6$ (both $4s$ electrons removed)

  • Iron(III) Cation: $\text{Fe}^{3+}\quad [\text{Ar}]\,3d^5$ (two $4s$ electrons and one $3d$ electron removed)

• Step-by-Step Transition Metal Practice Problems:

  • Problem 1: Determine the number of unpaired electrons in a Vanadium ion with a $2+$ charge ($\text{V}^{2+}$).

  • Step 1: Write the ground-state electron configuration of neutral Vanadium ($Z=23$):       $\text{V}: [\text{Ar}]\,4s^2 3d^3$

  • Step 2: Remove two electrons according to the highest-$n$ rule to form the $2+$ cation:       $\text{V}^{2+}: [\text{Ar}]\,3d^3$

  • Step 3: Apply Hund's rule to distribute the three $3d$ electrons among the five degenerate $d$ orbitals ($d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$, $d_{z^2}$). Each electron occupies an individual orbital with parallel spins:       $\uparrow \quad \uparrow \quad \uparrow \quad \underline{\phantom{\uparrow}} \quad \underline{\phantom{\uparrow}}$

  • Result: There are exactly 3 unpaired electrons.

  • Problem 2: Identify the metal ion with a $2+$ charge possessing the configuration $[\text{Xe}]\,4f^{14} 5d^{10}$.

  • Step 1: Tally the total number of electrons present in the cation:       $\text{Electrons} = 54\,(\text{Xe}) + 14\,(4f) + 10\,(5d) = 78\,\text{electrons}$

  • Step 2: Since the chemical species possesses a net charge of $2+$, the neutral atom must contain two additional protons and electrons:       $Z = 78 + 2 = 80$

  • Step 3: Locate atomic number $Z = 80$ on the periodic table, which corresponds to Mercury ($\text{Hg}$).

  • Step 4: Confirm ground-state neutral Mercury configuration: $[\text{Xe}]\,4f^{14} 5d^{10} 6s^2$. Removing the two outer $6s$ electrons gives $[\text{Xe}]\,4f^{14} 5d^{10}$.

  • Result: The metal ion symbol is $\text{Hg}^{2+}$.