Unit 4: Intermolecular Forces, Liquids & Solid-State Crystal Structures

Condensed matter forms when intermolecular attractive potential energies equal or exceed the thermal kinetic energy ($U_{attr} \ge k_B T$). While intramolecular covalent and ionic bonds ($\sim 150 - 1000\text{ kJ}\cdot\text{mol}^{-1}$) hold individual molecules together, the physical properties of liquids and solids—boiling points, vapor pressures, viscosities, and crystal morphologies—are dictated by intermolecular forces (van der Waals interactions and hydrogen bonds).

Classification of Intermolecular Interactions

Intermolecular interactions are primarily electromagnetic in origin, governed by Coulomb's law and quantum mechanical fluctuations:

1. Ion-Dipole Interactions ($V \propto r^{-2}$):

The electrostatic attraction between a free ion of charge $z e$ and a polar molecule with permanent electric dipole moment $\mu$:

This strong interaction ($\sim 40 - 200\text{ kJ}\cdot\text{mol}^{-1}$) drives the dissolution and hydration of ionic salts in polar solvents such as water (e.g., $\text{Na}^+(aq)$ surrounded by an octahedral primary hydration shell of 6 water molecules).

2. Dipole-Dipole Interactions (Keesom Forces, $V \propto r^{-6}$):

Between two rotating molecules possessing permanent dipole moments $\mu_1$ and $\mu_2$. When molecules undergo rapid thermal rotation at temperature $T$, Boltzmann angular weighting favors attractive orientations over repulsive ones. Averaging over all orientations yields the Keesom orientational potential (1912):

Because thermal collisions disrupt alignment, Keesom forces weaken inversely with temperature ($1/T$).

3. Dipole-Induced Dipole Interactions (Debye Forces, $V \propto r^{-6}$):

A permanent dipole $\mu_1$ induces an instantaneous polarization dipole in a neighboring nonpolar molecule of polarizability $\alpha_2$: $\mu_{ind} = \alpha_2 E$. Averaged over all orientations, Peter Debye (1920) derived the attractive potential:

Debye forces are temperature-independent because the induced dipole tracks the orientation of the inducing permanent dipole instantaneously.

4. London Dispersion Forces (Instantaneous Dipole-Induced Dipole, $V \propto r^{-6}$):

In 1930, Fritz London demonstrated using second-order quantum mechanical perturbation theory that even spherically symmetric nonpolar atoms (such as He, Ar, and $\text{CH}_4$) attract each other. Quantum zero-point fluctuations of the electron cloud create an ephemeral instantaneous dipole $\mu_{inst}(t) \sim e \delta r$, which generates an electric field that induces a synchronous dipole in adjacent atoms:

where $I_1, I_2$ are the first ionization energies of the interacting species and $\alpha_1, \alpha_2$ are their electronic polarizabilities.

• London dispersion forces are universal—present between all atoms and molecules without exception.
• Dispersion forces scale strongly with the total number of electrons and spatial volume of the electron cloud (polarizability $\alpha$). This explains the progression of boiling points down the noble gases ($\text{He}: 4.2\text{ K} \rightarrow \text{Xe}: 165\text{ K}$) and halogens ($\text{F}_2 < \text{Cl}_2 < \text{Br}_2 < \text{I}_2$).

5. Hydrogen Bonding ($\sim 10 - 40\text{ kJ}\cdot\text{mol}^{-1}$):

A specialized, highly directional non-covalent bond formed between an electronegative donor atom bearing a proton ($D-\text{H}$, where $D \in \{\text{F}, \text{O}, \text{N}\}$) and an electronegative acceptor atom $A$ possessing a non-bonding lone pair ($:A$). Modern quantum chemical calculations show that hydrogen bonding is a multifaceted hybrid phenomenon:

The charge-transfer component involves partial electron donation from the acceptor lone pair $n(A)$ into the antibonding $\sigma^*(D-\text{H})$ molecular orbital, which lengthens and weakens the covalent $D-\text{H}$ bond and produces the characteristic red-shift observed in infrared vibrational spectroscopy.

The Lennard-Jones 12-6 Pair Potential

The total potential energy between two neutral molecules is modeled by the Lennard-Jones 12-6 Potential:

• The $r^{-6}$ attractive term represents the sum of van der Waals attractions (Keesom + Debye + London).
• The $r^{-12}$ repulsive term models Pauli exchange repulsion between overlapping closed-shell electronic orbitals.
• $\epsilon$ is the depth of the potential energy well at the equilibrium separation $r_e = 2^{1/6}\sigma \approx 1.122\sigma$.
• $\sigma$ is the finite distance where $V(\sigma) = 0$.

The macroscopic manifestations of intermolecular forces in the liquid state are governed by fluid mechanics, thermodynamics, and interface science.

Surface Tension ($\gamma$) & Surface Free Energy

Molecules in the bulk interior of a liquid experience isotropic attractive van der Waals interactions in all directions ($\sum \mathbf{F}_{bulk} = 0$). In contrast, molecules residing at the liquid-gas interface experience a net inward cohesive pull toward the bulk liquid, because attractive interactions with gas molecules are negligibly weak.

To move a molecule from the interior to the surface requires working against these cohesive forces, endowing surface molecules with excess potential energy.

• Surface Free Energy: The reversible work required to expand the surface area of a liquid by an infinitesimal area $dA$ at constant temperature and pressure:

• Surface Tension: The force per unit length acting parallel to the surface and perpendicular to a line of unit length:

Because systems spontaneously minimize Gibbs free energy ($dG \le 0$), liquids spontaneously minimize their surface area for a given volume, explaining why free liquid droplets in microgravity adopt spherical geometries (the geometric shape with minimum surface-area-to-volume ratio).

The Young-Laplace Equation & Capillary Phenomena

Across any curved liquid-fluid interface, surface tension creates a pressure discontinuity:

The Young-Laplace Equation (1805): For an interface with principal radii of curvature $R_1$ and $R_2$, the pressure jump across the interface is:

For a spherical droplet or gas bubble of radius $R$ ($R_1 = R_2 = R$):

Capillary Action & Jurin's Law

When a narrow glass capillary of internal radius $r$ is immersed in a liquid:

• If the liquid wets the glass (contact angle $\theta < 90^\circ$, e.g., clean water on glass where $\theta \approx 0^\circ$), adhesive liquid-solid forces exceed cohesive liquid-liquid forces. The meniscus curves into a concave spherical cap of radius $R = r / \cos\theta$.

The pressure directly beneath the meniscus is reduced by $\Delta P = \frac{2\gamma}{R} = \frac{2\gamma\cos\theta}{r}$. Hydrostatic equilibrium is reached when the hydrostatic column weight $\rho g h$ balances this pressure deficit:

Liquid rises in the capillary until $h$ satisfies Jurin's Law.

• If the liquid does not wet the wall ($\theta > 90^\circ$, e.g., liquid mercury on glass where $\theta \approx 140^\circ$), cohesive forces dominate, creating a convex meniscus that depresses below the external liquid level ($h < 0$).

Liquid Viscosity & Poiseuille Flow

The dynamic viscosity $\eta$ of a liquid quantifies its resistance to shear flow. In laminar flow through a cylindrical capillary tube of length $L$ and radius $R$ driven by pressure drop $\Delta P$, the volumetric flow rate $Q = dV/dt$ is governed by the Hagen-Poiseuille Equation:

In the laboratory, relative viscosity is measured with an Ostwald viscometer, comparing the efflux time $t$ of a liquid with a reference standard (water) under gravity ($\Delta P = \rho g h$):

Unlike gases, whose viscosity increases with temperature ($\eta_{gas} \propto \sqrt{T}$), liquid viscosity decreases exponentially with temperature, described by the Arrhenius-Frenkel hole theory:

where $E_\eta$ is the activation energy required for a molecule to escape its intermolecular cage and jump into an adjacent vacant hole in the dynamic liquid lattice.

Water ($\text{H}_2\text{O}$) is the universal solvent of physical chemistry and biology. Its physical properties deviate wildly from those expected from its light molecular weight ($M = 18.015\text{ g}\cdot\text{mol}^{-1}$) by comparison with neighboring Group 16 hydrides ($\text{H}_2\text{S}, \text{H}_2\text{Se}, \text{H}_2\text{Te}$).

Molecular Geometry & Electronic Structure

The isolated water molecule possesses $C_{2v}$ point group symmetry:

• Two polar covalent $\text{O}-\text{H}$ bonds of length $r_e = 0.958\text{ \AA} = 95.8\text{ pm}$.
• Experimental bond angle $\angle\text{H-O-H} = 104.5^\circ$, compressed from the ideal tetrahedral angle ($109.47^\circ$) by the strong steric repulsion between the two non-bonding oxygen lone pairs.
• Net permanent dipole moment $\mu = 1.854\text{ Debye} = 6.18 \times 10^{-30}\text{ C}\cdot\text{m}$.
• Electronegativity difference ($\chi_O = 3.44$ vs $\chi_H = 2.20$) polarizes the bonds, placing partial charges $\delta- \approx -0.8e$ on oxygen and $\delta+ \approx +0.4e$ on each hydrogen.

Each water molecule possesses two hydrogen bond donors (the two protons) and two hydrogen bond acceptors (the two lone pairs on oxygen). This allows water to form an average of nearly four hydrogen bonds per molecule arranged in a nearly ideal 3D tetrahedral network.

Macroscopic Anomalies of Liquid Water

1. Unusually High Melting and Boiling Points:

Extrapolating boiling points of Group 16 hydrides ($\text{H}_2\text{Te}: -2^\circ\text{C}, \text{H}_2\text{Se}: -41^\circ\text{C}, \text{H}_2\text{S}: -60^\circ\text{C}$) suggests that water should boil at $-90^\circ\text{C}$ and exist as a gas at room temperature. Its actual boiling point ($100.0^\circ\text{C}$) reflects the substantial enthalpy required to break its extensive tetrahedral hydrogen-bonding network ($\Delta H_{vap}^\circ = 40.66\text{ kJ}\cdot\text{mol}^{-1}$).

2. The Maximum Density Anomaly at $3.98^\circ\text{C}$:

Virtually all normal liquids expand monotonically upon heating ($\alpha_P > 0$). Liquid water exhibits a density maximum at $T = 3.98^\circ\text{C} = 277.13\text{ K}$ ($\rho_{max} = 0.99997\text{ g}\cdot\text{cm}^{-3}$).

• In crystalline Ice Ih, water freezes into an open, hexagonal, cage-like clathrate network of coordination number 4, possessing a low density of $\rho_{ice} = 0.9167\text{ g}\cdot\text{cm}^{-3}$. Consequently, ice expands upon freezing and floats on water.
• Upon melting at $0^\circ\text{C}$, the rigid open cage partially collapses into disordered interstitial spaces, increasing density.
• Between $0^\circ\text{C}$ and $3.98^\circ\text{C}$, structural collapse of residual hydrogen-bonded clusters outpaces normal thermal expansion, yielding negative thermal expansion ($\alpha_P < 0$). Above $3.98^\circ\text{C}$, normal kinetic expansion dominates.
• This anomaly is vital for planetary life: aquatic lakes freeze from the top down, leaving liquid water at $4^\circ\text{C}$ insulated beneath ice layers.

3. High Dielectric Constant ($\epsilon_r = 78.4$ at $298\text{ K}$):

The cooperative orientation of hydrogen-bonded dipoles shields electrostatic charges by a factor of nearly 80. By Coulomb's law:

Attractive forces between $\text{Na}^+$ and $\text{Cl}^-$ ions in water are reduced by $98.7\%$, facilitating spontaneous ionic dissociation.

4. Colossal Specific Heat Capacity ($c_p = 4.184\text{ J}\cdot\text{g}^{-1}\cdot\text{K}^{-1}$):

Thermal energy supplied to water is absorbed into bending and breaking hydrogen bonds rather than immediately increasing translational kinetic energy, conferring an immense thermal buffer that stabilizes Earth's climate and cellular biology.

Crystalline solids are characterized by long-range three-dimensional periodic translational order. The mathematical framework describing crystals is crystallography.

Space Lattices, Basis & The 14 Bravais Lattices

• Space Lattice: An infinite, periodic, three-dimensional array of mathematical points in space, generated by discrete translation vectors:

where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are primitive basis translation vectors.

• Basis (Motif): The physical entity—an atom, molecule, or group of ions—attached identically to every lattice point:

• Unit Cell: The smallest repeating parallelopiped volume element whose translation through vectors $\mathbf{R}$ generates the entire macroscopic crystal. It is defined by six lattice parameters: three edge lengths $(a, b, c)$ and three interaxial angles $(\alpha, \beta, \gamma)$.

In 1848, Auguste Bravais proved mathematically that in three-dimensional space, there exist precisely 14 distinct translation lattices (Bravais Lattices) categorized into 7 Crystal Systems:

1. Cubic: $a = b = c, \alpha = \beta = \gamma = 90^\circ$ (Primitive P, Body-Centered I, Face-Centered F).

2. Tetragonal: $a = b \neq c, \alpha = \beta = \gamma = 90^\circ$ (Primitive P, Body-Centered I).

3. Orthorhombic: $a \neq b \neq c, \alpha = \beta = \gamma = 90^\circ$ (Primitive P, Body-Centered I, Face-Centered F, Base-Centered C).

4. Hexagonal: $a = b \neq c, \alpha = \beta = 90^\circ, \gamma = 120^\circ$ (Primitive P).

5. Rhombohedral (Trigonal): $a = b = c, \alpha = \beta = \gamma \neq 90^\circ$ (Primitive P).

6. Monoclinic: $a \neq b \neq c, \alpha = \gamma = 90^\circ, \beta \neq 90^\circ$ (Primitive P, Base-Centered C).

7. Triclinic: $a \neq b \neq c, \alpha \neq \beta \neq \gamma \neq 90^\circ$ (Primitive P).

The Three Cubic Unit Cells & Packing Efficiency

In physical chemistry, metallic elements and simple ionic salts predominantly crystallize in the Cubic System:

1. Simple Cubic (SC / Primitive P)

• Atom Coordinates: 8 corners at $(0,0,0)$ and lattice vectors.
• Atoms per Unit Cell: Each corner atom is shared among 8 adjacent unit cells:

• Coordination Number (CN): Each atom contacts 6 nearest neighbors along Cartesian axes ($\text{CN} = 6$).
• Lattice Parameter Relation: Atoms touch along cube edges:

• Atomic Packing Factor (APF):

The simple cubic structure is highly inefficient; only polonium ($\alpha\text{-Po}$) adopts this lattice.

2. Body-Centered Cubic (BCC / I)

• Atom Coordinates: 8 corners plus 1 atom at cube center $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$.
• Atoms per Unit Cell:

• Coordination Number: $\text{CN} = 8$ (central atom touches all 8 corner atoms).
• Lattice Parameter Relation: Atoms touch along the cube body diagonal of length $\sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$:

• Atomic Packing Factor (APF):

Examples: $\text{Fe}(\alpha), \text{Cr}, \text{W}, \text{Na}, \text{K}$.

3. Face-Centered Cubic (FCC / F / Cubic Close-Packed CCP)

• Atom Coordinates: 8 corners plus 6 face centers at $(\frac{1}{2}, \frac{1}{2}, 0)$, etc.
• Atoms per Unit Cell:

• Coordination Number: $\text{CN} = 12$ (maximum possible for identical hard spheres).
• Lattice Parameter Relation: Atoms touch along the face diagonal of length $\sqrt{a^2 + a^2} = a\sqrt{2}$:

• Atomic Packing Factor (APF):

Examples: $\text{Cu}, \text{Ag}, \text{Au}, \text{Al}, \text{Ni}, \text{Pt}$. FCC along with Hexagonal Close-Packed (HCP) ($\text{ABAB}\dots$ stacking sequence, also with $\text{CN} = 12$ and $\text{APF} = 74.05\%$, e.g., $\text{Mg}, \text{Ti}, \text{Zn}$) represents the closest possible sphere packing in 3D space (Kepler conjecture).

Bragg's Law of X-Ray Diffraction

In 1912, William Henry Bragg and William Lawrence Bragg demonstrated that crystalline planes act as three-dimensional diffraction gratings for monochromatic X-rays ($\lambda \sim 0.1\text{ nm}$).

Consider X-rays incident at glancing angle $\theta$ upon parallel crystallographic planes separated by interplanar spacing $d_{hkl}$:

• The path difference between waves reflected from adjacent planes is $\Delta = 2 d_{hkl} \sin\theta$.
• Constructive interference occurs when this path length equals an integer number $n$ of wavelengths:

For a cubic crystal with lattice constant $a$, the interplanar spacing $d_{hkl}$ for Miller indices $(hkl)$ is:

Solids are classified by the nature of the chemical forces binding their structural units together, while their mutual thermodynamic transformations are governed by the thermodynamics of phase equilibrium.

Major Classes of Crystalline Solids

1. Ionic Solids ($\text{NaCl}, \text{CsCl}, \text{CaF}_2, \text{ZnS}$):

• Structural Units: Positive cations and negative anions arranged in 3D alternating electrostatic lattices.
• Binding Forces: Non-directional Coulombic electrostatic attractions ($\Delta H_{latt} \sim 600 - 3000\text{ kJ/mol}$).
• Properties: High melting points, brittle (mechanical shear brings like-charged ions into repulsive contact, causing cleavage), electrical insulators in the solid state but excellent conductors when molten or dissolved in water.
• Radius Ratio Rule: The stable coordination geometry is dictated by the limiting radius ratio $\rho = r_+ / r_-$:
• $0.225 \le \rho < 0.414$: Tetrahedral ($\text{CN} = 4$, Zinc blende $\text{ZnS}$).
• $0.414 \le \rho < 0.732$: Octahedral ($\text{CN} = 6$, Rock salt $\text{NaCl}$).
• $0.732 \le \rho < 1.000$: Cubic ($\text{CN} = 8$, Caesium chloride $\text{CsCl}$).

2. Covalent Network Solids (Diamond, Graphite, Silicon, Quartz $\text{SiO}_2$):

• Structural Units: Atoms covalently bonded into infinite 2D or 3D macromolecular networks.
• Binding Forces: Directional covalent bonds ($\sim 350 - 500\text{ kJ/mol}$).
• Properties: Colossally high melting points (Diamond $T_m > 3800\text{ K}$), extreme hardness (diamond is Mohs 10), electrical insulators (diamond) or semimetals (graphite with delocalized $\pi$-electrons).

3. Metallic Solids ($\text{Fe}, \text{Cu}, \text{Ag}, \text{Na}$):

• Structural Units: Positive metal ion cores immersed in a delocalized, mobile "sea" of conduction valence electrons.
• Binding Forces: Metallic bonding (quantum mechanical band structure).
• Properties: High electrical and thermal conductivities ($\sigma \propto \tau$, Wiedemann-Franz law $\kappa / \sigma T = \text{const}$), ductile and malleable (non-directional electron sea accommodates lattice plane sliding without fracture).

4. Molecular Solids (Ice $\text{H}_2\text{O}$, Dry Ice $\text{CO}_2$, $\text{I}_2$, Sucrose):

• Structural Units: Discrete molecules held internally by strong covalent bonds.
• Binding Forces: Weak intermolecular van der Waals forces and hydrogen bonds ($\sim 1 - 40\text{ kJ/mol}$).
• Properties: Low melting points ($< 300^\circ\text{C}$), soft, electrical insulators.

5. Amorphous Solids & Glasses:

Substances lacking long-range three-dimensional translational periodicity (e.g., fused silica glass, amorphous polymers). They exhibit broad diffuse rings in X-ray diffraction, lack distinct melting points, and soften continuously across a glass transition temperature $T_g$.

The Clausius-Clapeyron Equation for Phase Equilibrium

Consider the phase equilibrium between a liquid and its vapor:

At equilibrium, the molar Gibbs free energies (chemical potentials) of both phases are identical:

Along the liquid-vapor coexistence curve, any infinitesimal variation $(dT, dP)$ that preserves equilibrium requires:

Rearranging gives the exact Clapeyron Equation:

For liquid-vapor equilibria:

1. Molar volume of vapor is colossally larger than molar volume of liquid: $\Delta V_{vap} = V_{m,vap} - V_{m,liq} \approx V_{m,vap}$.
2. Assuming the vapor behaves as an ideal gas: $V_{m,vap} = \frac{R T}{P}$.

Substituting these approximations into the Clapeyron equation:

Assuming the enthalpy of vaporization $\Delta H_{vap}$ is constant across a moderate temperature interval and integrating between $(T_1, P_1)$ and $(T_2, P_2)$ yields the integrated Clausius-Clapeyron Equation:

A plot of $\ln P$ versus $1/T$ is a straight line with slope $m = -\frac{\Delta H_{vap}}{R}$, providing the primary experimental method for determining enthalpies of vaporization.