6.3 Phase Diagrams & Critical Phenomena

6.3 Phase Diagrams & Critical Phenomena

Quick Summary: A pressure-temperature ($P-T$) phase diagram maps the thermodynamically stable phases of a pure substance. Regions (solid, liquid, gas) are delineated by coexistence curves representing two-phase equilibria. These curves meet at the invariant triple point, where three phases coexist. The vaporization curve ends at the critical point, beyond which liquid and gas merge into a supercritical fluid. While typical substances display a positive solid-liquid slope ($dP/dT > 0$), water displays an anomalous negative slope ($dP/dT < 0$) due to the open hexagonal hydrogen-bonded structure of ice.

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1. Topography of a Pressure-Temperature ($P-T$) Phase Diagram

A unary phase diagram displays the favored state of a pure substance across pressure ($P$) and temperature ($T$):

• Solid Region: Favored at low $T$ and high $P$ where IMFs dominate over thermal motion.
• Liquid Region: Occupies intermediate temperatures and moderate-to-high pressures.
• Gas Region: Favored at high $T$ and low $P$ where kinetic energy disperses molecules.

Coexistence Boundary Curves

Lines separating regions represent conditions where two phases coexist in dynamic equilibrium:

• Sublimation Curve: Solid $\rightleftharpoons$ Gas equilibrium. Originates near absolute zero and terminates at the triple point.
• Fusion Curve: Solid $\rightleftharpoons$ Liquid equilibrium. Extends upward from the triple point.
• Vaporization Curve: Liquid $\rightleftharpoons$ Gas equilibrium (the liquid vapor pressure curve). Extends from the triple point to the critical point.

Gibbs Phase Rule

Equilibrium constraints are governed by the Gibbs Phase Rule: $F = C - P_{phases} + 2$ For a pure substance ($C = 1$):

• Single-Phase Area ($P_{phases} = 1$): $F = 1 - 1 + 2 = 2$ (bivariant). Both $T$ and $P$ can vary independently.
• Coexistence Curve ($P_{phases} = 2$): $F = 1 - 2 + 2 = 1$ (univariant). Choosing $T$ automatically fixes $P$.
• Triple Point ($P_{phases} = 3$): $F = 1 - 3 + 2 = 0$ (invariant). Three phases coexist at a single fixed condition.

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2. Singular Thermodynamic Points: Triple Point and Critical Point

The Invariant Triple Point

The triple point ($T_{tp}, P_{tp}$) is the intersection where sublimation, fusion, and vaporization curves meet. At this invariant point, solid, liquid, and vapor coexist in simultaneous equilibrium:

• Water: $T_{tp} = 0.01^\circ\text{C}$ ($273.16\text{ K}$) and $P_{tp} = 0.00603\text{ atm}$ ($4.58\text{ torr} = 611.65\text{ Pa}$).
• Carbon Dioxide: $T_{tp} = -56.6^\circ\text{C}$ ($216.6\text{ K}$) and $P_{tp} = 5.11\text{ atm}$ ($518\text{ kPa}$).

The Critical Point

The vaporization curve terminates at the critical point ($T_c, P_c$):

• Critical Temperature ($T_c$): Maximum temperature at which a liquid can exist, regardless of applied pressure. Beyond $T_c$, kinetic energy exceeds all intermolecular attractions.
• Critical Pressure ($P_c$): Minimum pressure required to liquefy a gas at its critical temperature.

As liquid and vapor approach $(T_c, P_c)$ in a sealed cell, thermal expansion decreases liquid density while compression increases vapor density. At the critical point, both densities equalize, surface tension drops to zero, the meniscus vanishes, and the phases merge into a single fluid.

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3. Supercritical Fluids: Nature, Properties & Industrial Applications

A substance beyond its critical point ($T > T_c$ and $P > P_c$) is a supercritical fluid (SCF), exhibiting hybrid properties:

• Density: Liquid-like ($0.2 - 0.9\text{ g/cm}^3$), providing high solvent power for nonvolatile solutes.
• Viscosity & Diffusivity: Gas-like viscosity ($10^{-4} - 10^{-5}\text{ Pa}\cdot\text{s}$) and high diffusivity ($10^{-4}\text{ cm}^2/\text{s}$), enabling rapid mass transfer.
• Zero Surface Tension: Eliminates phase interfaces, permitting rapid penetration through microporous matrices.

Technological Applications

• Supercritical CO₂ ($\text{scCO}_2$): Accessible critical coordinates ($T_c = 31.0^\circ\text{C}$, $P_c = 72.8\text{ atm}$). Non-toxic, non-flammable, and leaves zero hazardous residues upon depressurization. Used to decaffeinate green coffee beans, extract botanical oils, and perform green dry cleaning.
• Supercritical Water ($\text{scH}_2\text{O}$): Critical coordinates ($T_c = 374^\circ\text{C}$, $P_c = 218\text{ atm}$). Collapsed hydrogen bonding lowers its dielectric constant, making it act as a nonpolar solvent. Used in Supercritical Water Oxidation (SCWO) to combust toxic organic wastes into CO₂, H₂O, and salts.

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4. Comparative Phase Topography: Normal $\text{CO}_2$ versus Anomalous $\text{H}_2\text{O}$

The slope of the fusion boundary is governed by the Clapeyron equation: $\frac{dP}{dT} = \frac{\Delta H_{fus}}{T (V_{liquid} - V_{solid})}$ Because melting is endothermic ($\Delta H_{fus} > 0$), the sign of $dP/dT$ depends entirely on $\Delta V_{fus} = V_{liquid} - V_{solid}$.

Carbon Dioxide (Typical Behavior)

• Solids are denser than liquids ($V_{liquid} > V_{solid} \implies \Delta V_{fus} > 0$).
• The fusion curve has a positive slope ($dP/dT > 0$, tilts right). Compressing liquid CO₂ causes it to freeze.
• Sublimation at 1 atm: The triple point is $5.11\text{ atm}$. Standard pressure ($1.00\text{ atm}$) lies far below the triple point. Therefore, liquid CO₂ cannot exist at $1\text{ atm}$. Solid dry ice sublimes directly to vapor at $-78.5^\circ\text{C}$. Liquid CO₂ requires pressures $\ge 5.11\text{ atm}$.

Water (Anomalous Behavior)

• Solid ice Ih has an open hexagonal hydrogen-bonded lattice with substantial void space. Upon melting, the framework partially collapses, packing molecules closer together. Liquid water is denser than solid ice at 0 °C ($\rho_{liq} = 1.000\text{ g/cm}^3$ vs $\rho_{ice} = 0.917\text{ g/cm}^3$).
• Thus, molar volume contracts: $\Delta V_{fus} = V_{liquid} - V_{solid} < 0$.
• By the Clapeyron equation, the fusion curve has an anomalous negative slope ($dP/dT < 0$, tilts left).
• Pressure-Induced Melting: Increasing pressure on ice slightly below 0 °C forces it to melt into the denser liquid phase without added heat. By Le Chatelier's principle, pressure favors the denser, lower-volume state.
• Phase Sequence at 1 atm: Water's triple point ($0.00603\text{ atm}$) lies far below $1.000\text{ atm}$. Heating ice at $1\text{ atm}$ cleanly traverses solid $\to$ liquid ($0.00^\circ\text{C}$) $\to$ gas ($100.00^\circ\text{C}$).

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5. Comparison Table of $\text{H}_2\text{O}$ and $\text{CO}_2$ Phase Boundaries