Chemical Bonds and Molecular Geometry

A covalent bond is the sharing of one or more pairs of electrons between two atoms (valence bond theory). Bonds can be single (one pair, e.g. $\mathrm{H_2}$), double (two pairs, e.g. $\mathrm{O_2}$) or triple (three pairs, e.g. $\mathrm{N_2}$). The Morse curve describes potential energy as a function of internuclear distance: the minimum corresponds to the bond length (equilibrium distance) and the well depth to the bond energy (how much energy is needed to break the bond).

Electronegativity ($\chi$) is the tendency of an atom to attract shared electrons in a bond, measured on the Pauling scale (0.7 to 4.0). The electronegativity difference $\Delta\chi$ between two atoms determines the bond type:

• $\Delta\chi < 0.4$: pure covalent (nonpolar) — electrons are shared equally (e.g. $\mathrm{H_2}, \mathrm{Cl_2}, \mathrm{CH_4}$)
• $0.4 \leq \Delta\chi < 1.7$: polar covalent — electrons are shifted toward the more electronegative atom, creating a dipole (e.g. $\mathrm{HCl}, \mathrm{H_2O}$)
• $\Delta\chi \geq 1.7$: ionic — the more electronegative atom completely strips away one or more electrons (e.g. $\mathrm{NaCl}$)

The dipole moment $\mu = q\cdot d$ measures the charge separation in a molecule. A molecule with polar bonds can be nonpolar if the individual dipoles cancel by symmetry (e.g. linear $\mathrm{CO_2}$), or polar if the resultant vector is nonzero (e.g. bent $\mathrm{H_2O}$).

Real-world applications: molecular polarity determines solubility ("like dissolves like": polar substances in polar solvents, nonpolar in nonpolar). Surfactants (soaps) have a polar hydrophilic head and a nonpolar hydrophobic tail, enabling grease emulsification in water.

The ionic bond forms by complete electron transfer from one atom to another: the atom that loses electrons becomes a cation (positive charge), the one that gains them becomes an anion (negative charge). The two ions are held together by electrostatic attraction (Coulomb's law). Ionic compounds are crystalline solids at room temperature with high melting points (e.g. NaCl melts at 801°C).

The lattice energy $U_{lat}$ is the energy released when gaseous ions assemble to form an ionic crystal. It is very large and negative (e.g. -788 kJ/mol for NaCl), and depends on:

• Ionic charges ($z^+$, $z^-$): higher charges → greater lattice energy (e.g. $\mathrm{MgO}$ with $\mathrm{Mg^{2+}}$ and $\mathrm{O^{2-}}$ has roughly 4× the lattice energy of NaCl)
• Interionic distance ($r_0$): smaller ions → shorter distance → stronger attraction
• Madelung constant ($M$): depends on the crystal lattice geometry (e.g. NaCl has a face-centered cubic structure with $M=1.7476$)

$$U_{lat} = -\dfrac{N_A M z^+ z^- e^2}{4\pi\varepsilon_0 r_0}\left(1-\dfrac{1}{n}\right)$$
Real-world applications: ionic compounds like NaCl (table salt), CaCO₃ (calcium carbonate, marble) and NaOH (sodium hydroxide) are ubiquitous. Electrical conductivity in the molten state or in solution arises from mobile ions (electrolytes), the principle behind batteries and electrolysis.

VSEPR theory (Valence Shell Electron Pair Repulsion) is based on a simple principle: electron pairs (both bonding and lone) in the valence shell of a central atom arrange themselves as far apart as possible to minimize electrostatic repulsion. The total number of electron domains (bonding pairs + lone pairs) determines the base geometry.

Hybridization explains how atomic orbitals combine to form equivalent bonding orbitals:

• $sp$ (2 domains) → linear geometry (180°), e.g. $\mathrm{CO_2}$, $\mathrm{BeCl_2}$, $\mathrm{C_2H_2}$ (acetylene)
• $sp^2$ (3 domains) → trigonal planar geometry (120°), e.g. $\mathrm{BF_3}$, $\mathrm{SO_3}$, $\mathrm{C_2H_4}$ (ethylene)
• $sp^3$ (4 domains) → tetrahedral geometry (109.5°), e.g. $\mathrm{CH_4}$, $\mathrm{CCl_4}$; with lone pairs the geometry distorts (e.g. $\mathrm{NH_3}$ pyramidal 107°, $\mathrm{H_2O}$ bent 104.5°)

Resonance: when a single Lewis structure is insufficient to describe a molecule, multiple resonance limit structures are used. The real structure is a resonance hybrid (average of the limit structures), with delocalized electrons. Classic examples:

• Benzene ($\mathrm{C_6H_6}$): 2 Kekulé structures, all C-C bonds are equivalent (order 1.5)
• Carbonate ion ($\mathrm{CO_3^{2-}}$): 3 resonance structures, each C-O bond has order 4/3

Real-world applications: molecular geometry determines biological properties: protein shape (tertiary structure) depends on bond angles, and drugs are designed to geometrically fit into receptors (lock-and-key).

The metallic bond is described by the "sea of electrons" model: metal atoms lose their valence electrons, which become delocalized and move freely among the metal cations arranged in a regular lattice. This delocalization explains the characteristic properties of metals:

• Electrical and thermal conductivity — free electrons carry charge and heat
• Ductility and malleability — ions can slide without breaking the metallic bond
• Luster — free electrons interact with light (reflection)

Band theory extends the model by describing continuous energy levels (bands) rather than discrete levels: the valence band (filled) and conduction band (empty or partially filled). The energy difference between them (band gap) determines whether a material is a conductor (zero gap, overlapping bands), semiconductor (small gap, 1-3 eV), or insulator (large gap, >5 eV).

Intermolecular forces (increasing strength):

• London (dispersion) forces: instantaneous dipoles from temporary fluctuations in the electron cloud. Universal (present in all molecules), increase with molecular mass and surface area. Example: noble gases liquefy only through these forces.
• Dipole-dipole (Van der Waals): attraction between the positive pole of one polar molecule and the negative pole of another. Stronger than London forces (for molecules of comparable mass).
• Hydrogen bond: the strongest dipole-dipole interaction, formed when hydrogen is bonded to a very electronegative atom (F, O, N). Explains water's anomalous properties (high boiling point, ice density < liquid water), the double helix structure of DNA (H-bonds between complementary bases), and protein structure.

Real-world applications: semiconductors like doped silicon (electronics), metal alloys (steel, brass), adhesives and lubricants, the molecular "Velcro" of receptor-drug interactions.