2.3 Chemical Bonding Models: Lewis Structures, VSEPR Theory, and Molecular Polarity

Chemical Bonding Models: Lewis Structures, VSEPR Theory, and Molecular Polarity

Chemical bonding lies at the heart of inorganic structure and reactivity. The PRC Chemist Licensure Examination tests candidates on both thermodynamic models of bonding (Born-Haber cycles, lattice energies, and the Kapustinskii equation) and structural models (Lewis structures, formal charges, resonance, VSEPR geometry predictions, and molecular symmetry).

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1. Ionic Bonding Energetics and Lattice Energy

An ionic bond consists of non-directional electrostatic attractions between oppositely charged spherical ions arranged in an extended crystalline lattice. The thermodynamic stability of an ionic solid is quantified by its lattice energy ($U_0$).

Definitions and Conventions

• Lattice Enthalpy of Formation ($\Delta H^\circ_{latt}$): Enthalpy change when 1 mole of an ionic crystalline solid is formed from its constituent gaseous ions (exothermic, $\Delta H^\circ_{latt} < 0$): $M^{z+}(g) + X^{z-}(g) \rightarrow MX(s) \quad \Delta H^\circ_{latt} < 0$
• Lattice Dissociation Energy ($U_0$): Energy required to separate 1 mole of solid into infinitely separated gaseous ions (endothermic, $U_0 = -\Delta H^\circ_{latt} > 0$).

The Born-Haber Thermochemical Cycle

Because lattice energy cannot be measured directly by experiment, it is calculated from accessible thermodynamic quantities using a Born-Haber cycle based on Hess's Law of constant heat summation:

$\Delta H^\circ_f = \Delta H^\circ_{sub}(M) + IE_1(M) + \frac{1}{2} \Delta H^\circ_{diss}(X_2) + \Delta H^\circ_{EA}(X) + \Delta H^\circ_{latt}$

Worked Thermodynamic Calculation: Sodium Chloride ($NaCl$) Given experimental standard enthalpies at $298\text{ K}$:

1. Standard enthalpy of formation of $NaCl(s)$: $\Delta H^\circ_f = -411.2\text{ kJ/mol}$
2. Enthalpy of sublimation of sodium: $Na(s) \rightarrow Na(g) \quad \Delta H^\circ_{sub} = +107.3\text{ kJ/mol}$
3. First ionization energy of sodium: $Na(g) \rightarrow Na^+(g) + e^- \quad IE_1 = +495.8\text{ kJ/mol}$
4. Bond dissociation enthalpy of chlorine: $\frac{1}{2} Cl_2(g) \rightarrow Cl(g) \quad \frac{1}{2} D(Cl_2) = \frac{1}{2}(+242.6) = +121.3\text{ kJ/mol}$
5. Electron affinity of chlorine: $Cl(g) + e^- \rightarrow Cl^-(g) \quad \Delta H^\circ_{EA} = -348.6\text{ kJ/mol}$

Applying Hess's Law: $-411.2 = 107.3 + 495.8 + 121.3 - 348.6 + \Delta H^\circ_{latt}$ $-411.2 = +375.8 + \Delta H^\circ_{latt}$ $\Delta H^\circ_{latt} = -411.2 - 375.8 = -787.0\text{ kJ/mol}$ Thus, the lattice dissociation energy of $NaCl$ is $U_0 = +787.0\text{ kJ/mol}$.

Theoretical Models: Born-Landé and Kapustinskii Equations

1. The Born-Landé Equation: Accounts for attractive Coulombic forces and short-range Born repulsive forces: $U_0 = \frac{N_A M |z_+ z_-| e^2}{4\pi \varepsilon_0 r_0} \left(1 - \frac{1}{n}\right)$ where $N_A$ is Avogadro's number, $M$ is the geometric Madelung constant ($1.7476$ for $NaCl$ rock-salt structure; $1.7627$ for $CsCl$; $1.6381$ for zinc blende; $1.641$ for wurtzite), $z_+$ and $z_-$ are ionic charge numbers, $r_0$ is equilibrium internuclear separation, and $n$ is the Born exponent ($5$ to $12$).

2. The Kapustinskii Equation: Enables calculation of lattice energy without knowing the crystal structure or Madelung constant by approximating $M$ as a function of the number of ions: $U_0 = \frac{1.202 \times 10^5 \cdot \nu \cdot |z_+ z_-|}{r_+ + r_-} \left(1 - \frac{34.5}{r_+ + r_-}\right) \quad \text{[kJ/mol]}$ where $\nu$ is the number of ions per formula unit (e.g., $\nu = 2$ for $NaCl$, $\nu = 3$ for $CaCl_2$) and ionic radii $r_+, r_-$ are entered in picometers ($\text{pm}$).

Worked Example: Comparing $NaCl$ vs $MgO$ ($r_{Mg^{2+}} = 72\text{ pm}, r_{O^{2-}} = 140\text{ pm}$, sum $= 212\text{ pm}$): $U_0(MgO) = \frac{1.202 \times 10^5 \cdot 2 \cdot (2 \times 2)}{212} \left(1 - \frac{34.5}{212}\right) = \frac{961600}{212} (1 - 0.1627) = 4535.8 \times 0.8373 \approx 3798\text{ kJ/mol}$ Because lattice energy scales with $|z_+ z_-|$, the quadrupled charge product in $MgO$ ($|(+2)(-2)| = 4$ versus $|(+1)(-1)| = 1$ in $NaCl$) increases lattice energy nearly five-fold, producing refractory behavior (melting point of $MgO = 2852^\circ\text{C}$ vs $NaCl = 801^\circ\text{C}$).

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2. Covalent Bonding, Formal Charges, and Modern Hypervalence

Formal Charge Calculation and Resonance

The formal charge ($FC$) represents the hypothetical electric charge assigned to an atom in a molecule, assuming shared bonding electrons are split equally between bonded atoms:

$FC = V - N - \frac{B}{2}$

where $V$ is valence electrons in the isolated neutral atom, $N$ is number of nonbonding valence electrons (lone pair electrons), and $B$ is total number of shared bonding electrons (so $B/2$ equals number of covalent bonds).

Rules for Selecting Optimal Resonance Contributors:

1. Structures with formal charges closest to zero are most stable.
2. Negative formal charges must reside on the most electronegative atoms.
3. Structures with adjacent like charges (e.g., $+1$ adjacent to $+1$) are severely destabilized.

Case Study: The Thiocyanate Ion ($[SCN]^-$): Total valence electrons $= 6(S) + 4(C) + 5(N) + 1(\text{charge}) = 16$.

• Form I: $[:\ddot{S}-C\equiv N:]^- \implies FC(S) = 6 - 6 - 1 = -1; \; FC(C) = 4 - 0 - 4 = 0; \; FC(N) = 5 - 2 - 3 = 0$
• Form II: $[:\ddot{S}=C=\ddot{N}:]^- \implies FC(S) = 6 - 4 - 2 = 0; \; FC(C) = 4 - 0 - 4 = 0; \; FC(N) = 5 - 4 - 2 = -1$
• Form III: $[:S\equiv C-\ddot{N}:^{2-}]^- \implies FC(S) = 6 - 2 - 3 = +1; \; FC(C) = 4 - 0 - 4 = 0; \; FC(N) = 5 - 6 - 1 = -2$ Evaluation: Form II is the major resonance contributor because Nitrogen is significantly more electronegative ($\chi = 3.04$) than Sulfur ($\chi = 2.58$). Placing the negative charge on Nitrogen (Form II) is energetically superior to Sulfur (Form I). Form III is negligible due to charge separation ($+1$ on S, $-2$ on N).

Modern Interpretation of Hypervalence (3c-4e Bonding)

Historically, Lewis octet expansion in Period 3+ elements ($PCl_5, SF_6, XeF_2$) was rationalized via Pauling's $sp^3d$ and $sp^3d^2$ hybridization invoking outer $d$-orbitals. However, modern quantum chemical calculations show that $3d$ orbitals in phosphorus and sulfur are far too high in energy and radially diffuse to engage in covalent bonding.

Instead, hypervalent bonding is explained by 3-center 4-electron (3c-4e) bonding (Pimentel-Rundle model):

• In linear $XeF_2$ ($AX_2E_3$), three equatorial lone pairs occupy $sp^2$-like hybrid orbitals in the equatorial plane.
• The axial $F-Xe-F$ linkage is formed by the linear overlap of one unhybridized $5p_z$ orbital on $Xe$ with a $2p_z$ orbital on each $F$ atom.
• This linear combination generates three molecular orbitals: one bonding ($\sigma$), one nonbonding ($\sigma_{nb}$), and one antibonding ($\sigma^*$): $\psi_b = \frac{1}{2} \phi_{F1} + \frac{1}{\sqrt{2}} \phi_{Xe} + \frac{1}{2} \phi_{F2}$ $\psi_{nb} = \frac{1}{\sqrt{2}} (\phi_{F1} - \phi_{F2}) \quad \text{[density localized entirely on fluorine atoms]}$
• Four electrons occupy $\psi_b$ and $\psi_{nb}$. Because the nonbonding orbital has nodes at Xenon, the net bonding electron pair is delocalized over two bonds, giving each $Xe-F$ bond a bond order of $0.5$ and placing significant partial negative charge on the electronegative Fluorine atoms without requiring $d$-orbital participation!

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3. Valence Shell Electron Pair Repulsion (VSEPR) Theory

VSEPR theory posits that valence electron pairs surrounding a central atom repel one another electrostatically and adopt a three-dimensional spatial orientation that maximizes their mutual angular separation.

Electron Pair Repulsion Hierarchy

$\text{Lone Pair - Lone Pair (lp-lp)} > \text{Lone Pair - Bonding Pair (lp-bp)} > \text{Bonding Pair - Bonding Pair (bp-bp)}$ Physical Rationale: A lone pair is electrostatically attracted to only one atomic nucleus; its electron cloud is broad, diffuse, and concentrated close to the central atom. In contrast, a bonding pair is stretched between two positive nuclei, confining its electron density to an internuclear cylinder. Consequently, lone pairs exert greater angular repulsive pressure, compressing adjacent bond angles:

• $CH_4$ ($AX_4$, no lone pairs): ideal tetrahedral angle $= 109.5^\circ$
• $NH_3$ ($AX_3E$, 1 lone pair): compressed to $107.3^\circ$
• $H_2O$ ($AX_2E_2$, 2 lone pairs): compressed to $104.5^\circ$

Steric Number (SN) and Complete Geometry Classification

The Steric Number is defined as: $SN = (\text{number of bonded atoms}) + (\text{number of lone pairs on central atom})$

The Steric Number 5 Equatorial Preference Rule

In a trigonal bipyramid, positions are non-equivalent:

• An axial position has three $90^\circ$ neighbors (high repulsive interaction).
• An equatorial position has only two $90^\circ$ neighbors and two $120^\circ$ neighbors.

Because electron repulsion decreases precipitously with angle (repulsion at $90^\circ \gg$ repulsion at $120^\circ$), lone pairs ALWAYS occupy equatorial positions in $SN=5$!

• 1 lone pair ($AX_4E$, e.g., $SF_4$): lone pair is equatorial $\implies$ Seesaw.
• 2 lone pairs ($AX_3E_2$, e.g., $ClF_3$): both lone pairs are equatorial $\implies$ T-shaped.
• 3 lone pairs ($AX_2E_3$, e.g., $XeF_2$): all three lone pairs reside in the equatorial plane, cancelling repulsions $\implies$ Linear ($180^\circ$).

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4. Molecular Polarity and Symmetry

A chemical bond between atoms of differing electronegativity possesses a bond dipole moment: $\mu = q \cdot r$ measured in Debye ($1\text{ D} = 3.336 \times 10^{-30}\text{ C}\cdot\text{m}$). The molecular dipole moment $\vec{\mu}_{net}$ is the vector sum of all individual bond dipoles and lone-pair dipole contributions: $\vec{\mu}_{net} = \sum_{i} \vec{\mu}_i$

A molecule has a net dipole moment (is polar, $\mu > 0$) only if its 3D geometry lacks symmetry elements that invert or cancel dipole vectors:

• Nonpolar Molecules ($\mu = 0$): High symmetry cancels all bond dipoles:
  • Linear $AX_2$ ($CO_2, BeCl_2, CS_2$)
  • Trigonal planar $AX_3$ ($BF_3, SO_3$)
  • Tetrahedral $AX_4$ ($CH_4, CCl_4$)
  • Trigonal bipyramidal $AX_5$ ($PCl_5$)
  • Linear $AX_2E_3$ ($XeF_2$)
  • Octahedral $AX_6$ ($SF_6$)
  • Square planar $AX_4E_2$ ($XeF_4$)
• Polar Molecules ($\mu > 0$): Asymmetric distributions of dipoles and lone pairs:
  • Bent $AX_2E$ ($SO_2, \mu = 1.63\text{ D}$)
  • Trigonal pyramidal $AX_3E$ ($NH_3, \mu = 1.47\text{ D}$; $PCl_3$ is also polar)
  • Bent $AX_2E_2$ ($H_2O, \mu = 1.85\text{ D}$; $OF_2, \mu = 0.30\text{ D}$)
  • Seesaw $AX_4E$ ($SF_4, \mu = 0.63\text{ D}$)
  • T-shaped $AX_3E_2$ ($ClF_3, \mu = 0.56\text{ D}$)
  • Square pyramidal $AX_5E$ ($BrF_5, \mu = 1.51\text{ D}$)