Chemical Bonding and Molecular Structure

In 1916, W. Kossel and G.N. Lewis independently proposed the electronic basis of chemical bonding. Atoms attain stability by acquiring an inert gas electronic configuration ($ns^2 np^6$, or $1s^2$ for Helium), known as the Octet Rule. This occurs via:

• Electrovalent (Ionic) Bonding: Complete transfer of one or more valence electrons from an electropositive atom to an electronegative atom (Kossel).
• Covalent Bonding: Mutual sharing of electron pairs between combining atoms with comparable electronegativities (Lewis).

While remarkably successful for second-period main group elements, the octet rule fails in three major classes of compounds:

1. Incomplete Octet of the Central Atom (Electron-Deficient Molecules): Central atoms with fewer than 8 valence electrons (e.g., $\text{LiCl}$ with 2, $\text{BeH}_2$ with 4, $\text{BF}_3$ and $\text{AlCl}_3$ with 6 valence electrons).
2. Odd-Electron Molecules: Molecules having an odd number of valence electrons where the octet cannot be satisfied for all atoms (e.g., Nitric oxide $\text{NO}$ with 11 valence electrons, Nitrogen dioxide $\text{NO}_2$ with 17 valence electrons). These species are paramagnetic.
3. Expanded Octet (Hypervalent Molecules): Elements of Period 3 and beyond have vacant $3d$ orbitals available for bonding and can accommodate 10, 12, or more valence electrons (e.g., $\text{PCl}_5$ with 10 electrons, $\text{SF}_6$ with 12 electrons, $\text{IF}_7$ with 14 electrons, $\text{H}_2\text{SO}_4$).

Formal charge is the hypothetical charge assigned to an individual atom in a Lewis polyatomic molecule or ion, assuming equal sharing of bonding electrons regardless of electronegativity differences:

Selection Rules for Optimal Lewis Structures:

• The most stable structure is the one where formal charges are closest to zero.
• Negative formal charges must reside on the most electronegative atoms.
• Structures with like formal charges on adjacent atoms are highly unfavorable.

The Lattice Enthalpy ($\Delta_L H^\circ$ or $U_L$) of an ionic solid is defined as the energy required to completely separate one mole of a solid ionic compound into its constituent gaseous ions at infinite distance:

Lattice enthalpy cannot be measured directly by experiment; it is calculated indirectly using Hess Law via the Born-Haber Cycle. For the synthesis of sodium chloride:

Factors Favoring Ionic Bond Formation: Low ionization enthalpy of the metallic element, high negative electron gain enthalpy of the non-metal, and high lattice enthalpy of the resulting crystal lattice.

In a homonuclear diatomic molecule ($\text{H}_2, \text{Cl}_2$), the bonding electron pair is shared equally, forming a non-polar covalent bond. In a heteronuclear molecule ($\text{HF}, \text{HCl}$), the more electronegative atom pulls the bonding pair closer, inducing partial charges ($\delta^+, \delta^-$) and creating a polar covalent bond.

The polarity of a molecule is quantitatively expressed by its Dipole Moment ($\vec{\mu}$):

Dipole moment is a vector quantity directed conventionally in chemistry from the electropositive atom to the electronegative atom ($\,+\hspace{-0.7em}\longrightarrow\,$).

In polyatomic molecules, the net molecular dipole moment is the vector sum of all individual bond dipoles and lone pair contributions:

• Symmetrical Polyatomics with Zero Dipole Moment:
  • $\text{CO}_2$ (Linear, $180^\circ$): The two equal and opposite $\text{C}=\text{O}$ bond dipoles cancel exactly: $\mu = 0\text{ D}$.
  • $\text{BF}_3$ (Trigonal planar, $120^\circ$): The vector resultant of any two $\text{B}-\text{F}$ dipoles is equal and opposite to the third: $\mu = 0\text{ D}$.
  • $\text{CCl}_4, \text{CH}_4$ (Regular tetrahedral, $109.5^\circ$): Tetrahedral symmetry cancels all four bond vectors: $\mu = 0\text{ D}$.
• The $\text{NH}_3$ vs $\text{NF}_3$ Dipole Paradox:

  Both ammonia ($\text{NH}_3$) and nitrogen trifluoride ($\text{NF}_3$) have pyramidal geometries with one lone pair on nitrogen. However, the dipole moment of $\text{NH}_3$ ($1.47\text{ D}$) is dramatically larger than that of $\text{NF}_3$ ($0.24\text{ D}$), despite fluorine being far more electronegative than hydrogen!

  Physical Explanation: In $\text{NH}_3$, nitrogen is more electronegative than hydrogen; the three $\text{N}-\text{H}$ bond dipoles point upward toward nitrogen, reinforcing the orbital dipole of the lone pair. In $\text{NF}_3$, fluorine is more electronegative than nitrogen; the three $\text{N}-\text{F}$ bond dipoles point downward, opposing and largely neutralizing the lone pair dipole.

No covalent bond between different atoms is 100% covalent, and no ionic bond is 100% ionic. The percentage ionic character is estimated from the electronegativity difference ($\Delta \chi = |\chi_A - \chi_B|$) using the Hannay-Smith Equation:

When $\Delta \chi = 1.7$, the bond is roughly 50% ionic and 50% covalent. When $\Delta \chi > 1.7$, the bond is predominantly ionic; when $\Delta \chi < 1.7$, it is predominantly covalent.

When a positive ion (cation) approaches a negative ion (anion), the cation attracts the anion electron cloud while repelling its nucleus. This distortion of the anion electron cloud is termed polarization. Polarization introduces electron sharing, imparting covalent character to an ionic bond.

Fajan Rules define the conditions favoring covalent character:

1. Small Cation Size: A smaller cation has a concentrated positive charge and high polarizing power (e.g., $\text{Li}^+ > \text{Na}^+ > \text{K}^+$; hence $\text{LiCl}$ is significantly more covalent than $\text{NaCl}$ and soluble in organic solvents).
2. Large Anion Size: A larger anion has a diffuse, loosely held valence electron cloud with high polarizability (e.g., $\text{I}^- > \text{Br}^- > \text{Cl}^- > \text{F}^-$; hence $\text{AgI}$ is yellow and highly covalent, while $\text{AgF}$ is ionic and water-soluble).
3. High Charge on Cation or Anion: Higher ionic charges multiply electrostatic distortion (e.g., $\text{SnCl}_4$ with $\text{Sn}^{4+}$ is a covalent liquid, whereas $\text{SnCl}_2$ with $\text{Sn}^{2+}$ is an ionic solid).
4. Electronic Configuration of the Cation (Pseudo-Noble Gas Configuration): Cations with an 18-electron valence shell ($ns^2 np^6 nd^{10}$, e.g., $\text{Cu}^+, \text{Ag}^+, \text{Zn}^{2+}$) have much greater polarizing power than cations of identical size and charge with an 8-electron noble gas shell ($ns^2 np^6$, e.g., $\text{Na}^+, \text{Ca}^{2+}$), because $d$-electrons provide poor nuclear shielding. Consequently, $\text{CuCl}$ is far more covalent and insoluble than $\text{NaCl}$.

The Valence Shell Electron Pair Repulsion (VSEPR) theory, developed by Sidgwick, Powell, Nyholm, and Gillespie, predicts 3D molecular geometry based on minimizing electrostatic repulsion between valence electron pairs surrounding the central atom:

1. The geometry of a molecule depends entirely on the total number of valence shell electron pairs (both bonding pairs, bp, and lone pairs, lp) around the central atom.
2. Electron pairs orient themselves in space at maximum distances to minimize electrostatic repulsion.
3. A lone pair occupies more spatial volume than a bonding pair because a lone pair is localized on a single nucleus, whereas a bonding pair is held electrostatically between two nuclei.
4. Hierarchy of Electron Pair Repulsions: $$\text{Lone Pair - Lone Pair (lp-lp)} > \text{Lone Pair - Bond Pair (lp-bp)} > \text{Bond Pair - Bond Pair (bp-bp)}$$
5. Multiple bonds (double or triple) are treated as single super-pairs in determining molecular geometry, though they exert greater repulsive force than single bonds.

The presence of lone pairs distorts ideal bond angles through enhanced $lp-bp$ repulsions:

• $AB_2E$ Type ($\text{SO}_2, \text{O}_3$): 2 bond pairs + 1 lone pair. Electron geometry is trigonal planar; molecular shape is Bent / V-shaped. Bond angle contracts from $120^\circ$ to $119.5^\circ$.
• $AB_3E$ Type ($\text{NH}_3, \text{PCl}_3$): 3 bond pairs + 1 lone pair. Electron geometry is tetrahedral; molecular shape is Trigonal Pyramidal. The $lp-bp$ repulsion compresses the $\text{H}-\text{N}-\text{H}$ angle from $109.5^\circ$ to $107^\circ$.
• $AB_2E_2$ Type ($\text{H}_2\text{O}, \text{H}_2\text{S}, \text{OF}_2$): 2 bond pairs + 2 lone pairs. Electron geometry is tetrahedral; molecular shape is Bent / Angular. The intense $lp-lp$ repulsion further squeezes the $\text{H}-\text{O}-\text{H}$ angle down to $104.5^\circ$.
• $AB_4E$ Type ($\text{SF}_4$): 4 bond pairs + 1 lone pair. Electron geometry is trigonal bipyramidal. The lone pair preferentially occupies an equatorial position (where it experiences only two $90^\circ$ repulsions, rather than three in an axial position). Molecular shape: See-saw.
• $AB_3E_2$ Type ($\text{ClF}_3, \text{BrF}_3$): 3 bond pairs + 2 lone pairs. Both lone pairs occupy equatorial sites to minimize repulsions. Molecular shape: T-shaped (bond angles $\sim 87.5^\circ$).
• $AB_2E_3$ Type ($\text{XeF}_2, \text{I}_3^-$): 2 bond pairs + 3 lone pairs. All three lone pairs occupy the equatorial plane symmetrically at $120^\circ$ to cancel repulsions. Molecular shape: Linear ($180^\circ$).
• $AB_5E$ Type ($\text{BrF}_5, \text{IF}_5$): 5 bond pairs + 1 lone pair. Electron geometry is octahedral; molecular shape: Square Pyramidal.
• $AB_4E_2$ Type ($\text{XeF}_4$): 4 bond pairs + 2 lone pairs. The two lone pairs occupy trans axial positions at $180^\circ$ to eliminate mutual repulsion. Molecular shape: Square Planar ($90^\circ$).

In trigonal bipyramidal $\text{PCl}_5$, the 5 $\text{P}-\text{Cl}$ bonds are not equivalent:

• The three equatorial bonds lie in one plane at $120^\circ$ to each other. Each equatorial bond experiences repulsive forces from only two axial bonds at $90^\circ$.
• The two axial bonds lie perpendicular above and below the equatorial plane. Each axial bond experiences repulsive forces from three equatorial bonds at $90^\circ$.

Due to greater repulsion, the axial bonds are longer and weaker than the equatorial bonds (Axial $\text{P}-\text{Cl} = 240\text{ pm}$, Equatorial $\text{P}-\text{Cl} = 202\text{ pm}$). Upon heating, $\text{PCl}_5$ readily dissociates by cleaving its weaker axial bonds: $\text{PCl}_5(g) \overset{\Delta}{\longrightarrow} \text{PCl}_3(g) + \text{Cl}_2(g)$.

Formulated by Heitler and London (1927) and extended by Linus Pauling, Valence Bond Theory describes a covalent bond as the pairing of electrons with anti-parallel spins through the overlap of half-filled atomic orbitals.

Energy Curve for Dihydrogen Formation: When two hydrogen atoms approach from infinity:

• Attractive forces (between nucleus of one atom and electron of the other) lower the potential energy.
• Repulsive forces (between both nuclei and between both electrons) raise the potential energy.
• At an equilibrium internuclear separation of $74\text{ pm}$ (the bond length), attractive forces balance repulsive forces, and potential energy reaches an absolute minimum of $-435.8\text{ kJ/mol}$ (the bond enthalpy). Closer approach causes nuclear repulsion to spike steeply.

The strength of a covalent bond depends directly on the extent of orbital overlap:

• Sigma ($\sigma$) Bond: Formed by end-to-end (coaxial or head-on) overlap of bonding orbitals along the internuclear axis.
  • $s-s$ overlap (e.g., $\text{H}_2$)
  • $s-p$ overlap (e.g., $\text{HF}, \text{HCl}$)
  • $p-p$ coaxial overlap (e.g., $\text{F}_2$)

  Because the overlap along the internuclear axis is maximal, a $\sigma$-bond is mechanically strong, has cylindrical electron density symmetry, and permits free rotation around the bond axis.

• Pi ($\pi$) Bond: Formed by sideways (lateral) overlap of two parallel $p$-orbitals perpendicular to the internuclear axis.

  The overlap consists of two saucer-shaped electron clouds above and below the internuclear plane. Because the extent of lateral overlap is considerably smaller than axial overlap, a $\pi$-bond is significantly weaker than a $\sigma$-bond. A $\pi$-bond never exists independently; it only forms in addition to a pre-existing $\sigma$-bond (e.g., double bond = $1\sigma + 1\pi$; triple bond = $1\sigma + 2\pi$). Rotation around a $\pi$-bond is restricted.

To explain why carbon forms four equivalent bonds in $\text{CH}_4$ with identical $109.5^\circ$ angles (despite having valence electrons in one spherical $2s$ and two mutually perpendicular $2p$ orbitals), Pauling introduced Hybridization: the phenomenon of intermixing atomic orbitals of slightly different energies to produce an entirely new set of equivalent orbitals with identical shapes, energies, and directional orientations.

• $sp$ Hybridization (Diagonal Hybridization):

  Mixing of one $s$ and one $p$ orbital $\longrightarrow$ two equivalent $sp$ hybrid orbitals oriented linearly at $180^\circ$ with 50% $s$-character and 50% $p$-character. Examples: $\text{BeCl}_2, \text{C}_2\text{H}_2$ (ethyne), $\text{CO}_2$.

• $sp^2$ Hybridization (Trigonal Hybridization):

  Mixing of one $s$ and two $p$ orbitals $\longrightarrow$ three equivalent $sp^2$ hybrid orbitals directed toward the vertices of an equilateral triangle at $120^\circ$ with 33.3% $s$-character. Examples: $\text{BF}_3, \text{BCl}_3, \text{C}_2\text{H}_4$ (ethene).

• $sp^3$ Hybridization (Tetrahedral Hybridization):

  Mixing of one $s$ and three $p$ orbitals $\longrightarrow$ four equivalent $sp^3$ hybrid orbitals directed toward the vertices of a regular tetrahedron at $109.5^\circ$ with 25% $s$-character. Examples: $\text{CH}_4, \text{NH}_3, \text{H}_2\text{O}, \text{CCl}_4$.

• $sp^3d$ Hybridization (Trigonal Bipyramidal):

  Mixing of one $s$, three $p$, and one $d_{z^2}$ orbital $\longrightarrow$ five $sp^3d$ hybrid orbitals. Three equatorial orbitals lie at $120^\circ$ and two axial orbitals lie at $90^\circ$. Example: $\text{PCl}_5$.

• $sp^3d^2$ Hybridization (Octahedral):

  Mixing of one $s$, three $p$, $d_{x^2-y^2}$, and $d_{z^2}$ orbitals $\longrightarrow$ six equivalent $sp^3d^2$ hybrid orbitals directed along the Cartesian axes at $90^\circ$. Example: $\text{SF}_6, [\text{SF}_6]$.

• $sp^3d^3$ Hybridization (Pentagonal Bipyramidal):

  Mixing of one $s$, three $p$, and three $d$ orbitals $\longrightarrow$ seven $sp^3d^3$ hybrid orbitals ($72^\circ$ equatorial, $90^\circ$ axial). Example: $\text{IF}_7$.

Developed by F. Hund and R.S. Mulliken (1932), Molecular Orbital Theory treats electrons in a molecule as belonging to the entire collection of nuclei (polycentric), rather than being localized between two atoms.

Linear Combination of Atomic Orbitals (LCAO) Approximation:

When two atomic orbital wavefunctions $\psi_A$ and $\psi_B$ combine:

• Bonding Molecular Orbital ($\psi_{\text{MO}} = \psi_A + \psi_B$): Constructive interference increases electron probability density between nuclei. Lower energy, greater stability than individual atomic orbitals. Designated as $\sigma, \pi$.
• Antibonding Molecular Orbital ($\psi^*_{\text{MO}} = \psi_A - \psi_B$): Destructive interference creates a nodal plane between nuclei where electron density drops to zero. Higher energy, destabilizing effect. Designated as $\sigma^*, \pi^*$.

Conditions for Combination of Atomic Orbitals: Combining AOs must possess comparable energies, identical symmetry with respect to the molecular axis, and maximum spatial overlap.

Electrons fill molecular orbitals following the Aufbau Principle, Pauli Exclusion Principle, and Hund Rule:

1. For Diatomics with 14 or Fewer Electrons ($\text{Li}_2, \text{Be}_2, \text{B}_2, \text{C}_2, \text{N}_2$):

   Strong $2s-2p_z$ orbital mixing destabilizes $\sigma 2p_z$, pushing it higher in energy than the degenerate $\pi 2p_x$ and $\pi 2p_y$ orbitals:

   $$\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < (\pi 2p_x = \pi 2p_y) < \sigma 2p_z < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$$
2. For Diatomics with More than 14 Electrons ($\text{O}_2, \text{F}_2, \text{Ne}_2$):

   The energy gap between $2s$ and $2p$ is large; minimal $s-p$ mixing occurs, and the normal expected energy order is preserved:

   $$\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < \sigma 2p_z < (\pi 2p_x = \pi 2p_y) < (\pi^* 2p_x = \pi^* 2p_y) < \sigma^* 2p_z$$

Physical Interpretations of Bond Order:

• $\text{BO} > 0$: The molecule is stable and capable of existence.
• $\text{BO} \le 0$: The molecule is unstable and cannot exist (e.g., $\text{He}_2$ has $N_b = 2, N_a = 2 \implies \text{BO} = 0$).
• $\text{BO} = 1$ corresponds to a single bond; $\text{BO} = 2$ to a double bond; $\text{BO} = 3$ to a triple bond. Fractional bond orders ($0.5, 1.5, 2.5$) represent resonance-stabilized or transient species.
• Correlations: Bond Enthalpy $\propto$ Bond Order; Bond Length $\propto \frac{1}{\text{Bond Order}}$.

Classical Lewis structures depict dioxygen with a complete octet and paired electrons ($\text{O}=\text{O}$), predicting diamagnetism. Experimentally, liquid oxygen is strongly attracted by a magnetic field (paramagnetic).

MOT Explanation for $\text{O}_2$ (16 electrons):

Here, $N_b = 10, N_a = 6 \implies \text{BO} = \frac{10-6}{2} = 2.0$. By Hund rule, the two highest-energy electrons enter the degenerate antibonding orbitals singly with parallel spins ($\pi^* 2p_x^1, \pi^* 2p_y^1$). These two unpaired electrons unambiguously account for the observed paramagnetism of oxygen!

Stability & Bond Strength Order: $\text{O}_2^+ > \text{O}_2 > \text{O}_2^- > \text{O}_2^{2-}$

Bond Length Order: $\text{O}_2^+ < \text{O}_2 < \text{O}_2^- < \text{O}_2^{2-}$

A Hydrogen Bond (H-bond) is the electrostatic attractive force between a hydrogen atom covalently bonded to a highly electronegative atom and another electronegative atom in the same or an adjacent molecule:

Essential Prerequisites for H-Bonding:

1. The hydrogen atom must be covalently attached to a very small, strongly electronegative atom (primarily Fluorine, Oxygen, or Nitrogen; occasionally Chlorine or $sp$-hybridized Carbon in $\text{HCN}$/alkynes under specific conditions).
2. The electronegative atom $\text{Y}$ must possess at least one lone pair of electrons to interact with the electropositive hydrogen.

The energy of a hydrogen bond ranges from $10$ to $40\text{ kJ/mol}$, making it much weaker than a true covalent bond ($200-450\text{ kJ/mol}$) but significantly stronger than ordinary van der Waals forces ($2-10\text{ kJ/mol}$).

• Intermolecular Hydrogen Bonding: Formed between different molecules of the same or different compounds.
  • $\text{H}_2\text{O}$ and $\text{HF}$: Extensive 3D hydrogen bonding networks cause molecular association.
  • Consequences: Abnormally elevated boiling points, higher viscosity, higher surface tension, and high solubility of alcohols and sugars in water.
• Intramolecular Hydrogen Bonding (Chelation): Formed within the same molecule between a hydrogen atom and an electronegative atom located on adjacent functional groups in close proximity.
  • Examples: ortho-Nitrophenol, Salicylaldehyde, Salicylic acid.
  • Consequences: Prevents molecular association with surrounding molecules; lowers boiling point and melting point; enhances volatility and solubility in non-polar organic solvents.

1. The Boiling Point Inversion of Group 15, 16, 17 Hydrides:

   Normally, boiling points increase down a group as molecular mass and London dispersion forces increase (e.g., $\text{H}_2\text{S} < \text{H}_2\text{Se} < \text{H}_2\text{Te}$). However, the first members—$\text{H}_2\text{O} (100^\circ\text{C})$, $\text{HF} (19.5^\circ\text{C})$, and $\text{NH}_3 (-33^\circ\text{C})$—display drastically higher boiling points than their heavier congeners due to powerful intermolecular hydrogen bonding that requires large thermal energy to break.

   Between $\text{H}_2\text{O}$ and $\text{HF}$, although $\text{H}-\text{F}$ forms stronger individual H-bonds due to higher electronegativity of fluorine, water boils higher ($100^\circ\text{C}$ vs $19.5^\circ\text{C}$) because each $\text{H}_2\text{O}$ molecule can form an average of 4 hydrogen bonds (2 hydrogens and 2 lone pairs), creating an extensive 3D network, whereas each $\text{HF}$ forms only 2 hydrogen bonds per molecule.

2. The Open-Cage Hexagonal Lattice of Ice and Density Anomaly:

   In ice, each oxygen atom is tetrahedrally surrounded by four other oxygen atoms via two covalent $\text{O}-\text{H}$ bonds and two intermolecular $\text{O}\cdots\text{H}$ hydrogen bonds at distances of $100\text{ pm}$ and $176\text{ pm}$ respectively. This tetrahedral geometry generates an open, cage-like hexagonal framework containing large interstitial voids.

   When ice melts at $0^\circ\text{C}$, the rigid cage collapses partially, and water molecules tumble into the vacant cavities. Consequently, liquid water packs more tightly than ice, giving ice a lower density ($\sim 0.917\text{ g/cm}^3$) than liquid water ($1.000\text{ g/cm}^3$) at $0^\circ\text{C}$. Water reaches its maximum density at $3.98^\circ\text{C}$ ($4^\circ\text{C}$), ensuring aquatic ecosystems survive under frozen lake surfaces in winter.

3. Separation of ortho-Nitrophenol and para-Nitrophenol:

   ortho-Nitrophenol undergoes intramolecular H-bonding (chelation) between its $-\text{OH}$ and $-\text{NO}_2$ groups, forming a stable six-membered ring. This internal bonding prevents association with other molecules, making it steam-volatile with a lower boiling point ($216^\circ\text{C}$). In contrast, para-Nitrophenol forms extensive intermolecular H-bonds linking molecules into long chains; it is non-steam-volatile with a much higher boiling point ($279^\circ\text{C}$). This enables clean separation by simple steam distillation.